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AN  ELEMENTAEY  TREATISE 


ON 


THE    DIFFERENTIAL    CALCULUS 


FOURTH   EDITION. 


AN  ELEMENTARY  TREATISE 

ON 

THE     INTEGRAL     CALCULUS, 

CONTAINING 

APPLICATIONS  TO  PLANE  CURVES  AND  SURFACES. 

BY 

BENJAMIN  WILLIAMSON,  F.R.S. 


IN    THE    PRESS. 


AN    ELEMENTARY  TREATISE   ON    DYNAMICS. 

BY 

BENJAMIN  WILLIAMSON,    F.E.S., 

AND 

FRANCIS   A.   TARLETON,    LL.  D., 

Fellows  of  Trinity  College,  Dublin. 


AN   ELEMENTARY  TREATISE 


ON 


THE  DIFFERENTIAL  CALCULUS, 


CONTAINING 


THE  THEORY  OF  PLANE  CURVES, 


WITH 


NUMEROUS    EXAMPLES. 


BY 
BENJAMIN  WILLIAMSON,  M.A.,  F.R.S., 

FELLOW  OF  TRINITY  COLLEGE,  AND  PROFESSOR  OF  NATURAL  PHILOSOPHY 
IN  THE  UNIVERSITY  OF  DUBLIN. 


NEW  YORK: 
APPLETON    AND    COMPANY 

1884. 


[all  rights  reserved.] 


9A50+ 

ixri 


242141 


BOSTON  COLLEGE  LIBRARY 
CHESTNUT  HILL,  MASS. 


PREFACE. 


In  the  following  Treatise  I  have  adopted  the  method  of 
Limiting  Eatios  as  my  basis ;  at  the  same  time  the  co- 
ordinate method  of  Infinitesimals  or  Differentials  has  been 
largely  employed.  In  this  latter  respect  I  have  followed  in 
the  steps  of  all  the  great  writers  on  the  Calculus,  from 
Newton  and  Leibnitz,  its  inventors,  down  to  Bertrand,  the 
author  of  the  latest  great  treatise  on  the  subject.  An  ex- 
clusive adherence  to  the  method  of  Differential  Coefficients 
is  by  no  means  necessary  for  clearness  and  simplicity  ;  and, 
indeed,  I  have  found  by  experience  that  many  fundamental 
investigations  in  Mechanics  and  Greometry  are  made  more 
intelligible  to  beginners  by  the  method  of  Differentials  than 
by  that  of  Differential  Coefficients.  While  in  the  more  ad- 
vanced applications  of  the  Calculus,  which  we  find  in  such 
works  as  the  Mecanique  Celeste  of  Laplace,  and  the  Meea- 
nique  Analytique  of  Lagrange,  the  investigations  are  all 
conducted  on  the  method  of  Infinitesimals.  The  principles 
on  which  this  method  is  founded  are  given  in  a  concise  form 
in  Arts.  38  and  39. 

In  the  portion  of  the  Book  devoted  to  the  discussion  of 
Curves,  I  have  not  confined  myself  exclusively  to  the  ap- 
plication of  the  Differential  Calculus  to  the  subject ;  but 
have  availed  myself  of  the  methods  of  Pure  and  Analytic 


vi  Preface. 

Greometry,  whenever  it  appeared  that  simplicity  would  be 
gained  thereby. 

In  the  discussion  of  Multiple  Points  I  have  adopted  the 
simple  and  Greneral  Method  given  by  Dr.  Salmon  in  his 
Higher  Plane  Curves.  It  is  hoped  that  by  this  means  the 
present  treatise  will  be  found  to  be  a  useful  introduction  to 
the  more  complete  investigations  contained  in  that  work. 

As  this  Book  is  principally  intended  for  the  use  of  begin- 
ners, I  have  purposely  omitted  all  metaphysical  discussions, 
from  a  conviction  that  they  are  more  calculated  to  perplex 
the  beginner  than  to  assist  him  in  forming  clear  conceptions. 
The  student  of  the  Differential  Calculus  (or  of  any  other 
branch  of  Mathematics)  cannot  expect  to  master  at  once  all 
the  difficulties  which  meet  him  at  the  outset ;  indeed  it  is  only 
after  considerable  acquaintance  with  the  Science  of  Greometry 
that  correct  notions  of  angles,  areas,  and  ratios  are  formed. 
Such  notions  in  any  science  can  be  acquired  only  after 
practice  in  the  application  of  its  principles,  and  after  patient 
study. 

The  more  advanced  student  may  read  with  advantage  the 
Reflexions  sur  la  Metaphysique  clu  Calcul  Infinitesimal  of  the 
illustrious  Oarnot :  in  which,  after  giving  a  complete  resume 
of  the  different  points  of  view  under  which  the  principles  of 
the  Calculus  may  be  regarded,  he  concludes  as  follows : — 

"  Le  merite  essentiel,  le  sublime,  on  peut  le  dire,  de  la 
methode  infinitesimale,  est  de  reunir  la  facilite  des  procedes 
ordinaires  d'un  simple  calcul  d' approximation  a  1' exactitude 
des  resultats  de  l'analyse  ordinaire.  Cet  avantage  immense 
serait  perdu,  ou  du  moins  fort  diminue,  si  a  cette  methode 
pure  et  simple,  telle  que  nous  l'a  donnee  Leibnitz,  on  voulait, 
sous  l'apparence  d'une  plus  grande  rigueur  soutenue  dans 
tout  le  cours  de  calcul,  en  substituer  d'autres  moins  naturelles, 


Preface.  vii 

moins  commodes,  moins  conformes  a  la  marche   probable 
des  inventeurs.     Si  cette  methode  est  exacte  dans  les  re- 
sultats,  comme  personne  n'en  doute  aujourd'hui,  si  c'est  tou- 
jours  a  elle  qu'il  faut  en  revenir  dans  les  questions  difficiles, 
comme  il  parait  encore  que  tout  le   monde  en  convient, 
pourquoi  recourir  a  des  moyens  detournes  et  compliques  pour 
la  suppleer?     Pourquoi  se  contenter  de  l'appuyer  sur  des 
inductions  et  sur  la  conformite  de  ses  resultats  avec  ceux  que 
fournissent  les  autres  methodes,  lorsqu'on  peut  la  demontrer 
directement    et    generalement,    plus    facilement    peut-etre 
qu'aucune  de  ces  methodes  elles-memes  ?    Les  objections  que 
Ton  a  faites  contre  elle  portent  toutes  sur  cette  fausse  suppo- 
sition, que  les  erreurs  commises  dans  le  cours  du  calcul,  en  y 
negligeant  les  quantites  infiniment  petites,  sont  demeurees 
dans  le  resultat  de  ce  calcul,  quelque  petites  qu'on  les  sup- 
pose; or  c'est  ce  qui  n'est  point:  1' elimination  les  emporte 
toutes   necessairement,   et  il   est  singulier   qu'on   n'ait  pas 
apercu  d'abord  dans  cette  condition  indispensable  de  1' elimi- 
nation le  veritable  caractere  des  quantites  infinitesimales  et 
la  reponse  dirimante  a  toutes  les  objections/' 

Many  important  portions  of  the  Calculus  have  been 
omitted,  as  being  of  too  advanced  a  character;  however, 
within  the  limits  proposed,  I  have  endeavoured  to  make  the 
"Work  as  complete  as  the  nature  of  an  elementary  treatise 
would  allow. 

I  have  illustrated  each  principle  throughout  by  copious 
examples,  chiefly  selected  from  the  Papers  set  at  the  various 
Examinations  in  Trinity  College. 

In  the  Chapter  on  Eoulettes,  in  addition  to  the  discussion 
of  Cycloids  and  Epicycloids,  I  have  given  a  tolerably  com- 
plete treatment  of  the  question  of  the  Curvature  of  a  Roulette, 
as  also  that  of  the  Envelope  of  any  Curve  carried  by  a  rolling 


viii  Preface. 

Curve.  This  discussion  is  based  on  the  beautiful  and  general 
results  known  as  Savary's  Theorems ;  taken  in  conjunction 
with  the  properties  of  the  Circle  of  Inflexions.  I  have  also 
introduced  the  application  of  these  theorems  to  the  general 
ease  of  the  motion  of  any  plane  area  supposed  to  move  on 
a  fixed  Plane. 

In  this  Edition  I  have  made  little  alteration  beyond  the 
introduction  of  a  short  account  of  the  principles  of  the  deter- 
minant functions  known  under  the  name  of  Jacobians,  which 
now  hold  so  fundamental  a  place  in  analysis. 

Tklntty  College, 
June,  1884. 


TABLE   OF    CONTENTS. 


CHAPTEB  I. 


FIKST   PRINCIPLES.       DIFFERENTIATION. 

Page 

Dependent  and  Independent  Variables, -  I 

Increments,  Differentials,  Limiting  Ratios,  Derived  Functions,          .        .  3 

Differential  Coefficients, .5 

Geometrical  Illustration,    ...........  6 

Navier,  on  the  Fundamental  Principles  of  the  Differential  Calculus,  .         .  8 

On  Limits, 10 

Differentiation  of  a  Product,       .         .         .         •         .         .         .         .  13 

Differentiation  of  a  Quotient,      .         .         .         .         .         .         .         •  15 

Differentiation  of  a  Power, 16 

Differentiation  of  a  Function  of  a  Function, 17 

Differentiation  of  Circular  Functions, 19 

Geometrical  Illustration  of  Differentiation  of  Circular  Functions,       .         .  22 

Differentiation  of  a  Logarithm, 24 

Differentiation  of  an  Exponential,      ...         .....  26 

Logarithmic  Differentiation, 27 

Examples, 30 


CHAPTER  II. 


SUCCESSIVE   DIFFERENTIATION. 


Successive  Differential  Coefficients,   . 
Infinitesimals,  ....... 

Geometrical  Illustrations  of  Infinitesimals, 
Fundamental  Principle  of  the  Infinitesimal  Calculus, 

Subsidiary  Principle, 

Approximations,        ...... 

Derived  Functions  of  xm,  .... 

Differential  Coefficients  of  an  Exponential, 

Differential  Coefficients  of  tan_1#,  and  tan-1-,    . 

x 

Theorem  of  Leibnitz, 

Applications  of  Leibnitz's  Theorem, 
Examples,         ....... 


34 
36 

37 

40 

4i 
42 
46 
48 

50 

Si 

53 

57 


Table  of  Contents. 


CHAPTER  III. 


DEVELOPMENT    OF   FuTSTCTIOlSrS. 


Taylor's  Expansion, 

Binomial  Theorem,  .  .... 

Logarithmic  Series,  .         .          .... 

Maclaurin's  Theorem,       ...... 

Exponential  Series, 

Expansions  of  sin  x  and  cos  x,  . 

Huygens'  Approximation  to  Length  of  Circular  Arc, 
Expansions  of  tan-1  x  and  sin-1  x,       .... 
Enler's  Expressions  for  sin  x  and  cos  x,     . 

John  Bernoulli's  Series, 

Symbolic  Eorm  of  Taylor's  Series,    .... 

Convergent  and  Divergent  Series,     .... 

Lagrange's  Theorem  on  the  Limits  of  Taylor's  Series, 

Geometrical  Illustration,  ..... 

Second  Form  of  the  Bemainder,        .... 

General  Form  of  Maclaurin's  Series, 

Binomial  Theorem  for  Fractional  and  Negative  Indices, 

Expansions  by  aid  of  Differential  Equations,     . 

Expansion  of  sin  mz  and  cos  mz,        .... 

Arbogast's  Method  of  Derivation,     . 

Examples,       ........ 


Page 
6l 

63 
63 
64 

65 
66 
66 
68 
69 
70 
70 

73 
76 
78 

79 
81 
82 

85 
87 


CHAPTER  IY. 


INDETERMINATE   FORMS. 

Examples  of  Evaluating  Indeterminate  Forms  without  the  Differential  Cal 

cuius,        .... 
Method  of  Differential  Calculus, 
Form  o  x  00  , 

Form^,         .... 

CO  ' 

Forms  o°,  00  °,  i±cc    . 
Examples,        .... 

CHAPTER  Y. 


PARTIAL  DIFFERENTIAL  COEFFICIENTS. 

Partial  Differentiation,     ....... 

Total  Differentiation  of  a  Function  of  Two  Variables, 

Total  Differentiation  of  a  Function  of  Three  or  more  Variables, 

Differentiation  of  a  Function  of  Differences, 

Implicit  Functions,  Differentiation  of  an  Implicit  Function, 

Euler's  Theorem  of  Homogeneous  Functions,    . 

Examples  in  Plane  Trigonometry, 

Landen's  Transformation,         .  .... 

Examples  in  Spherical  Trigonometry,        .... 
Legendre's  Theorem  on  the  Comparison  of  Elliptic  Functions, 
Examples,        ......... 


Table  of  Contents. 


XI 


CHAPTER  VI. 


SUCCESSIVE   PARTIAL   DIFFERENTIATION. 

The  Order  of  Differentiation  is  indifferent  in  Independent  Variables, 
Condition  that  Pdx  +  Qdy  should  he  an  exact  Differential, 
Euler's  Theorem  of  Homogeneous  Functions,           .... 
Successive  Differential  Coefficients  of  <p(x  +  at,  y  +  fit), 
Examples, 


Page 

H5 

146 

148 
148 
150 


CHAPTER  VII. 

lagrange's  theorem. 

Lagrange's  Theorem, 151 

Laplace's  Theorem,        .         .         .         .         .         .         .         .         .         .     154 

Examples,     ............     155 

CHAPTER  VIII. 


EXTENSION  OE  TAYLOR  S  THEOREM. 


Expansion  of  <p  (x  +  h,  y  +  7c), 
Expansion  of  <p  (x  +  h,  y  -f  k,  z  +  I), 
Symbolic  Forms,  .... 
Euler's  Theorem,  .... 


156 

J59 
160 

162 


CHAPTER  IX. 


MAXIMA    AND    MINIMA   POR   A   SINGLE  VARIABLE. 

Geometrical  Examples  of  Maxima  and  Minima, 

Algebraic  Examples,      ..... 

Criterion  for  a  Maximum  or  a  Minimum, 

Maxima  and  Minima  occur  alternately,  . 

Maxima  or  Minima  of  a  Quadratic  Fraction,    . 

Maximum  or  Minimum  Section  of  a  Eight  Cone, 

Maxima  or  Minima  of  an  Implicit  Function,  . 

Maximum  or  Minimum  of  a  Function  of  Two  Dependent  Variables, 

Examples,      ........... 


164 
165 
169 

173 

177 

181 
185 
186 


CHAPTER  X. 

MAXIMA  OR  MINIMA  OP  FUNCTIONS  OP  TWO  OR  MORE  VARIABLES. 

Maxima  and  Minima  for  Two  Variables,  ...... 

Lagrange's  Condition  in  the  case  of  Two  Independent  Variables,     . 


191 
191 


Xll 


Table  of  Contents. 


Maximum  or  Minimum  of  a  Quadratic  Fraction,     .... 

Application  to  Surfaces  of  Second  Degree, 

Maxima  and  Minima  for  Three  Variables,        .... 
Lagrange's  Conditions  in  the  case  of  Three  Variables, 
Maximum  or  Minimum  of  a  Quadratic  Function  of  Three  Variables, 
Examples, 


Page 
194 
196 
198 
199 
200 
203 


CHAPTER  XI. 

METHOD   OF   UNDETERMINED   MULTIPLIERS   APPLIED   TO    MAXIMA   AND 

MINIMA. 

Method  of  Undetermined  Multipliers, 204 

Application  to  find  the  principal  Eadii  of  Curvature  on  a  Surface,    .         .     208 
Examples,      ............     210 

CHAPTER  XII. 


ON   TANGENTS   AND    NORMALS    TO    CURVES. 

Equation  of  Tangent, 

Equation  of  Normal,  ..... 
Subtangent  and  Subnormal,  .... 
Number  of  Tangents  from  an  External  Point, 
Number  of  Normals  passing  through  a  given  Point, 

Differential  of  an  Arc, 

Angle  between  Tangent  and  Eaclius  Vector, 
Polar  Subtangent  and  Subnormal, 
Inverse  Curves,      ..... 
Pedal  Curves,         ..... 

Reciprocal  Polars, 

Pedal  and  Reciprocal  Polar  of  rm  =  am  cos  md, 

Intercept  between  point  of  Contact  and  foot  of  Perpendicular, 

Direction  of  Tangent  and  Normal  in  Vectorial  Coordinates, 

Symmetrical  Curves,  and  Central  Curves, 

Examples,      .  ........ 


212 
215 

215 
219 

220 

220 

222 

223 
225 
227 
228 
230 
232 

233 

236 

28 


CHAPTEE  XIII. 


ASYMPTOTES. 

Points  of  Intersection  of  a  Curve  and  a  Eight  Line, 
Method  of  Finding  Asymptotes  in  Cartesian  Coordinates, 
Case  where  Asymptotes  all  pass  through  the  Origin, 
Asymptotes  Parallel  to  Coordinate  Axes, 
Parabolic  and  Hyperbolic  Branches,        .... 

Parallel  Asymptotes, 

The  Points  in  which  a  Cubic  is  cut  by  its  Asymptotes  lie  in  a  Eight  Line 

Asymptotes  in  Polar  Curves, 

Asymptotic  Circles, 

Examples,   ......... 


240 
242 
245 
245 
246 
247 

249 
250 
252 

254 


Table  of  Contents.  xiii 


CHAPTER  XIY. 

MULTIPLE   POINTS    ON   CURVES. 

Page 

Nodes,  Cusps,  Conjugate  Points,  .        .        .  .        .        .  259 

Method  of  Finding  Double  Points  in  general, 261 

Parabolas  of  the  Third  Degree, 262 

Double  Points  on  a  Cubic  having  three  given  lines  for  its  Asymptotes,     .  264 

Multiple  Points  of  higher  Orders, 265 

Cusps,  in  general,  ..........  266 

Multiple  Points  on  Curves  in  Polar  Coordinates,      .         .         .         .         .267 

Examples,     .  268 

CHAPTER  XV. 

ENVELOPES. 

Method  of  Envelopes, 270 

Envelope  of  La?  +  2 Ma  +  iV=o, 272 

Uundetermined  Multipliers  applied  to  Envelopes,    .         .         .         .         -273 

Examples, 276 

CHAPTER  XVI. 

CONVEXITY,    CONCAVITY,    POINTS   OP   INFLEXION. 

Convexity  and  Concavity, 278 

Points  of  Inflexion, 279 

Harmonic  Polar  of  a  Point  of  Inflexion  on  a  Cubic,         .         .         .         .281 

Stationary  Tangents,     .........  282 

Examples, 283 

CHAPTER  XVII. 

RADIUS  OP  CURVATURE,  EVOLUTES,  CONTACT. 

Curvature,  Angle  of  Contingence, 285 

Eadius  of  Curvature, 286 

Expressions  for  Eadius  of  Curvature, 287 

Newton's  Method  of  considering  Curvature, 291 

Radii  of  Curvature  of  Inverse  Curves, 295 

Radius  and  Chord  of  Curvature  in  terms  of  r  and  p,        .         .         .         .  295 

Chord  of  Curvature  through  Origin,         .  296 

Evolutes  and  Involutes, 297 

Evolute  of  Parabola, 298 

Evolute  of  Ellipse, 299 

Evolute  of  Equiangular  Spiral, 300 


XIV 


Table  of  Contents. 


Page 

Involute  of  a  Circle, 3°° 

Eadius  of  Curvature  and  Points  of  Inflexion  in  Polar  Coordinates,            .  301 

Intrinsic  Equation  of  a  Curve, 3°4 

Contact  of  Different  Orders, 3°4 

Centre  of  Curvature  of  an  Ellipse, 3°7 

Osculating  Curves, 3°9 

Eadii  of  Curvature  at  a  Node, 3IQ 

Eadii  of  Curvature  at  a  Cusp,         ...                  ....  311 

At  a  Cusp  of  the  Second  Species  the  two  Eadii  of  Curvature  are  equal,    .  312 

General  Discussion  of  Cusps, 3X5 

Points  on  Evolute  corresponding  to  Cusps  on  Curve,        .         .         •         •  316 

Equation  of  Osculating  Conic, 3X7 

Examples, 3*9 


CHAPTEE  XYIII. 


ON"     TEACING     OP     CTJEVES. 


Tracing  Algebraic  Curves, 

Cubic  with  three  real  Asymptotes, 

Each  Asymptote  corresponds  to  two  Infinite  Branches, 

Tracing  Curves  in  Polar  Coordinates, 

On  the  Curves  rm  =  am  cos  m6,        .... 

The  Limacon, 

The  Conchoid,        ....... 

Examples, 


322 
323 
325 
328 
328 

331 
332 

333 


CHAPTEE  XIX. 


ROULETTES. 


Eoulettes,  Cycloid, 

Tangent  to  Cycloid, 

Eadius  of  Curvature,  Evolute, 

Length  of  Cycloid, 

Trochoids,      .         .         •         • 

Epicycloids  and  Hypocycloids, 

Eadius  of  Curvature  of  Epicycloid, 

Double  Generation  of  Epicycloids  and  Hypocycloids, 

Evolute  of  Epicycloid,   . 

Pedal  of  Epicycloid, 

Epitrochoid  and  Hypotrochoid,     _  . 

Centre  of  Curvature  of  Epitrochoid, 

Savary's  Theorem  on  Centre  of  Curvature  of  a  Eoulette, 

Geometrical  Construction  for  Centre  of  Curvature, 

Circle  of  Inflexions,        ...... 

Envelope  of  a  Carried  Curve,  .... 

Centre  of  Curvature  of  the  Envelope, 


335 
336 
337 
338 
339 
339 
342 
343 
344 
346 
347 
35i 
352 
352 
354 
355 
357 


XVI 


Table  of  Contents, 


Linear  Transformations  for  Three  Variables, 

Case  of  Orthogonal  Transformations, 

General  Case  of  Transformation  for  Two  Independent  Variables, 

Functions  unaltered  by  Linear  Transformations, 

Application  to  Geometry  of  Two  Dimensions, 

Application  to  Orthogonal  Transformations,    .... 

Jacobians,      .......... 

Case  in  which  Functions  are  not  Independent, 

Jacobian  of  Implicit  Functions, 

Case  where  J  =  o, 

Examples,     .......... 

Miscellaneous  Examples, 

Note  on  Failure  of  Taylor's  Theorem,      .... 
Note  on  General  Conditions  for  a  Maximum  or  Minimum, 


Page 
408 
409 
410 
411 
412 
414 

415 
417 
420 
422 
427 

430 

443 
445 


The  beginner  is  recommended  to  omit  the  following  portions  on  the  first 
reading :— Arts.  49,  50,  51,  52,  67-85,  88,  in,  114-116,  124,  125,  Chap,  til, 
Chap.  Yin.,  Arts.  159-163,  249-254,  261-269,  296-301,  Chap.  xxn. 


CORRIGENDA. 


-~         r     ..           „    n(n-i)..(n—r—i)                   ,  n(n—i) . ..  (n-r+i) 
Page  63,  line  9,  for  — — -xn-ryri  read  — ' 


1  . 2  ...r 


1 . 2  . ..  r 


xn~ryr. 


, ,     165,  third  line  from  bottom,/or  -  ±  a/ u,  read  -  ±  A 


,.           „    m  +  n  _  m  +  n 

213,  line  2,  for y,  read 


m 


,,     228,  second  line  from  bottom,  for  Art.  180,  read  Art.  176. 
,,     301,  fourth  line  from  bottom,  for  Art.  185,  read  Art.  183. 


Table  of  Contents. 


xv 


Eadius  of  Curvature  of  Envelope  of  a  Eight  Line, 

On  the  Motion  of  a  Plane  Figure  in  its  Plane, 

Chasles'  Method  of  Drawing  Normals,    . 

Motion  of  a  Plane  Figure  reduced  to  Eoulettes, 

Epicyclics,    . 

Properties  of  Circle  of  Inflexions, 

Theorem  of  Bobilier, 

Centre  of  Curvature  of  Conchoid, 

Spherical  Eoulettes, 

Examples,     .... 


Page 
358 

359 
360 
362 

363 
367 
368 
37o 
37o 
372 


CHAPTER  XX. 

ON   THE   CARTESIAN   OVAL. 


Equation  of  Cartesian  Oval,  . 
Construction  for  Third  Focus, 
Equation,  referred  to  each  pair  of  Foci, 
Conjugate  Ovals  are  Inverse  Curves, 
Construction  for  Tangent, 
Confocal  Curves  cut  Orthogonally, 
Cartesian  Oval  as  an  Envelope, 
Examples,     ..... 


375 
376 
377 
378 
379 
381 
382 

384 


CHAPTER  XXI. 


ELIMINATION    OF   CONSTANTS   AND    FUNCTIONS. 


Elimination  of  Constants, 384 

Elimination  of  Transcendental  Functions, 386 

Elimination  of  Arbitrary  Functions,  .  .  _  .  .  .  .  .387 
Condition  that  one  expression  should  be  a  Function  of  another,  .  .  389 
Elimination  in  the  case  of  Arbitrary  Functions  of  the  same  expression,  .  393 
Examples, •  397 


CHAPTER  XXII. 

CHANGE    OE   INDEPENDENT   VARIABLE. 

Case  of  a  Single  Independent  Variable,  .         .         . 
Transformation  from  Eectangular  to  Polar  Coordinates,  . 

d%V     d2V 

Transformation  of  -— —  +  -~rj> 

(too       ™y* 

p  dW     dW     dW 
Transformation  <&—+—+  —  ,.         .         .         . 

Geometrical  Illustration  of  Partial  Differentiation, 


399 
403 

404 

405 
407 


DIFFEKENTIAL  CALCULUS 


CHAPTEK  I. 

FIRST  PRINCIPLES — DIFFERENTIATION . 

i .  Functions. — The  student,  from  his  previous  acquaintance 
with  Algebra  and  Trigonometry,  is  supposed  to  understand 
what  is  meant  when  one  quantity  is  said  to  be  a  function  of 
another.  Thus,  in  trigonometry,  the  sine,  cosine,  tangent,  &c, 
of  an  angle  are  said  to  be  functions  of  the  angle,  having  each 
a  single  value  if  the  angle  is  given,  and  varying  when  the 
angle  varies.  In  like  manner  any  algebraic  expression  in  x 
is  said  to  be  a  function  of  x.  Geometry  also  furnishes  us 
with  simple  illustrations.  For  instance,  the  area  of  a  square, 
or  of  any  regular  polygon  of  a  given  number  of  sides,  is  a 
function  of  its  side  ;  and  the  volume  of  a  sphere,  of  its  radius. 

In  general,  whenever  two  quantities  are  so  related,  that 
any  change  made  in  the  one  produces  a  corresponding  variation 
in  the  other,  then  the  latter  is  said  to  be  a  function  of  the 
former. 

This  relation  between  two  quantities  is  usually  represented 
by  the  letters  F,  f,  <p,  &c. 

Thus  the  equations 

u  =  F(x),    v=f{x),    w  =  $(x), 

denote  that  u,  v,  w,  are  regarded  as  functions  of  x,  whose 
values  are  determined  for  any  particular  value  of  x,  when  the 
form  of  the  function  is  known. 

2.  Dependent  and  Independent  Variables,  Con- 
stants.— In  each  of  the  preceding  expressions,  x  is  said  to  be 

B 


2  First  Principles — Differentiation. 

the  independent  variable,  to  which  any  value  may  be  assigned 
at  pleasure ;  and  u,  v,  w,  are  called  dependent  variables,  as  their 
values  depend  on  that  of  x,  and  are  determined  when  it  is 
known. 

Thus,  in  the  equations 

y  =  10%     y  =  x3,     y  =  sin#, 

the  value  of  y  depends  on  that  of  x,  and  is  in  each  case  deter- 
mined when  the  value  of  x  is  given. 

If  we  suppose  any  series  of  values,  positive  or  negative, 
assigned  to  the  independent  variable  x,  then  every  function 
of  x  will  assume  a  corresponding  series  of  values.  If  a  quan- 
tity retain  the  same  value,  whatever  change  be  given  to  x,  it 
is  said  to  be  a  constant  with  respect  to  x.  We  usually  denote 
constants  by  a,  b,  c,  &c,  the  first  letters  of  the  alphabet ; 
variables  by  the  last,  viz.,  u,  v,  w,  x,  y,  z. 

3.  Algebraic  and  Transcendental  Functions. — 
Functions  which  consist  of  a  finite  number  of  terms,  involving 
integral  and  fractional  powers  of  x,  together  with  constants 
solely,  are  called  algebraic  functions — thus 


{x~a)"'    WT^f    («  +  *)(»-.*H**. 

are  algebraic  expressions. 

Functions  which  do  not  admit  of  being  represented  as 
ordinary  algebraic  expressions  in  a  finite  number  of  terms  are 
called  transcendental :  thus,  sin  x,  cos  x,  tan  x,  ex,  log  x,  &c, 
are  transcendental  functions ;  for  they  cannot  be  expressed 
in  terms  of  x  except  by  a  series  containing  an  infinite  number 
of  terms. 

Algebraic  functions  are  ultimately  reducible  to  the  follow- 
ing elementary  forms  :  (1).  Sum,  or  difference  (u  +  v,  u  -  v). 

(2).  Product,  and  its  inverse,  quotient  [uv,  -  J.    Powers,  and 

their  inverse,  roots  (um,  um). 

The  elementary  transcendental  functions  are  also  ulti- 
mately reducible  to :  (1).  The  sine,  and  its  inverse,  (sin  u, 
sin*1^).  (2).  The  exponential,  and  its  inverse,  logarithm 
(eu,  log  u). 


Limiting  Ratios — Derived  Functions.  3 

4.  Continuous  Functions. — A  function  <j>  (cc)  is  said  to 
be  a  continuous  function  of  x,  between  the  limits  a  and  b, 
when,  to  each  value  of  x,  between  these  limits,  corresponds  a 
finite  value  of  the  function,  and  when  an  infinitely  small 
change  in  the  value  of  x  produces  only  an  infinitely  small 
change  in  the  function.  If  these  conditions  be  not  fulfilled 
the  function  is  discontinuous.  It  is  easily  seen  that  all 
algebraic  expressions,  such  as 

&qX     t  Oi\X        t    •    •    •    .    dny 

and  all  circular  expressions,  sin  x,  tan  x,  &c,  are,  in  general, 
continuous  functions,  as  also  e°,  log  x,  &c.  In  such  cases, 
accordingly,  it  follows  that  if  x  receive  a  very  small  change, 
the  corresponding  change  in  the  function  of  x  is  also  \&ery 
small. 

5.  Increments  and  Differentials. — In  the  Differen- 
tial Calculus  we  investigate  the  changes  which  any  function 
undergoes  when  the  variable  on  which  it  depends  is  made  to 
pass  through  a  series  of  different  stages  of  magnitude. 

If  the  variable  x  be  supposed  to  receive  any  change,  such 
change  is  called  an  increment ;  this  increment  of  x  is  usually 
represented  by  the  notation  Ax. 

When  the  increment,  or  difference,  is  supposed  infinitely 
small  it  is  called  a  differential,  and  represented  by  dx,  i.e.  an 
infinitely  small  difference  is  called  a  differential. 

In  like  manner,  if  u  be  a  function  of  x,  and  x  becomes 
x  +  Ax,  the  corresponding  value  of  u  is  represented  by  u  +  Au ; 
i.  e.  the  increment  of  u  is  denoted  by  Au. 

6.  limiting  Ratios,  Derived  Functions. — If  u  be  a 
function  of  x,  then  for  finite  increments,  it  is  obvious  that  the 
ratio  of  the  increment  of  u  to  the  corresponding  increment  of 
x  has,  in  general,  a  finite  value.  Also  when  the  increment 
of  x  is  regarded  as  being  infinitely  small,  we  assume  that  the 
ratio  above  mentioned  has  still  a  definite  limiting  value.  In 
the  Differential  Calculus  we  investigate  the  values  of  these 
limiting  ratios  for  different  forms  of  functions. 

The  ratio  of  the  increment  of  u  to  that  of  x  in  the  limit, 

dif 
when  both  are  infinitely  small,  is  denoted  by  — .     When 

ax 

B  2 


4  First  Principles — Differentiation. 

u  =/(#),  this  limiting  ratio  is  denoted  by  /'(#),  and  is  called 
the  first  derived  function*  off(x). 

Thus ;  let  x  "become  x  +  h,  where  h  =  Ax,  then  u  becomes 

f(x  +  h),  i.e.u  +  Au  =f(%  +  h), 

.'.  Au  =f(x  +  h)  -f{x), 

Au     f(x  +  h)  -fix) 
Ax  h 

The  limiting  value  of  this  expression  when  h  is  infinitely  small 
is  called  the  first  derived  function  of  f(%),  and  represented 

Au 
Again,  since  the  ratio  —  has/'  (x)  for  its  limiting  value, 

i^X 

if  we  assume 

Au 
e  must  become  evanescent  along  with  Ax ;  also  —  becomes 

Ax 

—  at  the  same  time ;  hence  we  have 
dx 

This  result  may  be  stated  otherwise,  thus  : — If  ux  denote 
the  value  of  u  when  x  becomes  al9  then  the  value  of  the  ratio 

— ,  when  a?i  -  x  is  evanescent,  is  called  the  first  derived 

Xi  ~~  x 

du 
function  of  u,  and  denoted  by  — . 


*  The  method  of  derived  functions  was  introduced  by  Lagrange,  and  the 
different  derived  functions  of/  (x)  were  defined  by  him  to  be,  the  coefficients  of 
the  powers  of  h  in  the  expansion  of /(#  +  h) :  that  this  definition  of  the  first 
derived  function  agrees  with  that  given  in  the  text  will  be  seen  subsequently. 

This  agreement  was  also  pointed  out  by  Lagrange.  See  "Theorie  des 
Fonctions  Analytiques,"  Nos.  3,  9. 


Algebraic  Illustration.  5 

If  xL  be  greater  than  x,  then  ux  is  also  greater  than  u,  pro- 
vided — is  positive ;  and  hence,  in  the  limit,  when  xx  -  x 

Xi  -  x  .  du 

is  evanescent,  ux  is  greater  or  less  than  u  according  as  —  is 

ax 

positive  or  negative.  Hence,  if  we  suppose  x  to  increase, 
then  any  function  of  x  increases  or  diminishes  at  the  same 
time,  according  as  its  derived  function,  taken  with  respect 
to  x,  is  positive  or  negative.  This  principle  is  of  great 
importance  in  tracing  the  different  stages  of  a  function  of  x, 
corresponding  to  a  series  of  values  of  x. 

7.  Differential,    and   Differential   Coefficient,    of 

Let  u  =f(x) ;  then  since 

we  have  du  =  d{f(x))  =  f '  (x)  dx, 

where  dx  is  regarded  as  being  infinitely  small.  In  this 
case  dx  is,  as  already  stated,  the  differential  of  x,  and  du 
or  f  (x)  dx,  is  called  the  corresponding  differential  of  u. 
Also  f  (x)  is  called  the  differential  coefficient  of  f(x),  being 
the  coefficient  of  dx  in  the  differential  of  f{x). 

8.  Algebraic  Illustration. — That  a  fraction  whose 
numerator  and  denominator  are  both  evanescent,  or  in- 
finitely small,  may  have    a    finite    determinate   value,   is 

a       na 
evident  from  algebra.     For  example,  we  have  T  =  —  what- 

0      no 

ever  n  may  be.      If  n  be  regarded  as  an  infinitely  small 

number,  the  numerator  and  denominator  of  the  fraction 

both  become  infinitely  small  magnitudes,  while  their  ratio 

remains  unaltered  and  equal  to  -. 

It  will  be  observed  that  this  agrees  with  our  ordinary 
idea  of  a  ratio ;  for  the  value  of  a  ratio  depends  on  the 
relative,  and  not  on  the  absolute  magnitude  of  the  terms 
which  compose  it. 


Again,  ff  u  = 


na  +  n2a' 


nb  -r  n2tf' 
in  which  n  is  regarded  as  infinitely  small,  and  a,  b,  a'  and  b' 


6  First  Principles — Differentiation. 

represent  finite  magnitudes,  the  terms  of  the  fraction  are 

both  infinitely  small, 

i    j  j  i    •        j»«  Co  \  n& 

but  tneir  ratio  is -„ 

b  +  no 

the  limiting  value  of  which,  as  n  is  diminished  indefinitely, 

.     a 

is  -.      Again,  it  we  suppose  n  indefinitely  increased,   the 

limiting  value  of  the  fraction  is  y>.     For 
a  +  dn       d        ab'  -  bd 


b  +  b'n       V      bf(b  +  b'n)  ' 

but  the  fraction   77-77 — — — ■    diminishes    indefinitely  as   n 

b  (b  +  bn)  J 

increases   indefinitely,   and   may  be   made    less   than   any 
assignable    magnitude,    however    small.      Accordingly  the 

limiting  value  of  the  fraction  in  this  case  is  —.. 

b 

9.  Trigonometrical  Illustration. — To  find  the  values 
of  7 — 77,  and  — 77-,  when  0  is  regarded  as  infinitely  small. 

Here  - — ^  =  cos0,  and  when  0  =  o,  cos0  =  1. 
tan  u 

Hence,  in  the  limit,  when  9  =  o,*  we  have 

sin0  _         tan0  .  .. 

7 — 77  =  1,  and,  .*.  —. — jz  =  1,  at  the  same  time. 

tan  6  sin  6 

Again,  to  find  the  value  of  -r— 5,  when  0  is  infinitely  small. 

From  geometrical  considerations  it  is  evident  that  if  6  be 
the  circular  measure  of  an  angle,  we  have 

tan  0  >  6  >  sin  0, 

tan0        0 
or  -7— 7j  >  -t—x  >  1  ; 

sin  0      sin  0 

*  If  a  variable  quantity  be  supposed  to  diminish  gradually,  till  it  be  less  than 
anything  finite  which  can  be  assigned,  it  is  said  in  that  state  to  be  indefinitely 
small  or  evanescent;  for  abbreviation,  such  a  quantity  is  often  denoted  by  cypher. 

A  discussion  of  infinitesimals,  or  infinitely  small  quantities  of  different  orders, 
will  be  found  in  the  next  Chapter. 


Geometrical  Illustration. 

but  in  the  limit,  i.e.  when  0  is  infinitely  small, 

tan0 
"sin~0  =  *' 

and  therefore,  at  the  same  time,  we  have 

e 


sin0 


i. 


This  shows  that  in  a  circle  the  ultimate  ratio  of  an  arc  to  its 
chord  is  unity,  when  they  are  both  regarded  as  evanescent. 

10.  Geometrical  Illustration. — Assuming  that  the 
relation  y  =  f(x)  may  in  all  cases  be  represented  by  a  curve, 
where  ,  . 

expresses  the  equation  connecting  the  co-ordinates  (x,  y) 
of  each  of  its  points ;  then,  if  the  axes  be  rectangular,  and 
two  points  (x,  y),  (xx,  yi)  be  taken  on  the  curve,  it  is  obvious 

that  — represents  the  tangent  of  the  angle  which  the 

Xi  -  x  °  ° 

chord  joining  the  points  (a?,  y),  (xx,  y^)  makes  with  the  axis 
of  x. 

If,  now,  we  suppose  the  points  taken  infinitely  near  to 
each  other,  so  that  xx-  x  becomes  evanescent,  then  the  chord 
becomes  the  tangent  at  the  point  (x,  y),  but 

— — -  becomes  -f-  or  f  (x)  in  this  case. 

Hence,  f  (x)  represents  the  trigonometrical  tangent  of  the 
angle  which  the  line  touching  the  curve  at  the  point  (x,  y)  makes 
with  the  axis  of  x.  We  see,  accordingly,  that  to  draw  the 
tangent  at  any  point  to  the  curve 

y  =  f(x) 

is  the  same  as  to  find  the  derived  function  fix)  of  y  with 
respect  to  x.  Hence,  also,  the  equation  of  the  tangent  to 
the  curve  at  a  point  (x,  y)  is  evidently 

y-Y  -/(«)  (x-X),  (2) 

where  X,  Y  are  the  current  co-ordinates  of  any  point  on  the 


8 


First  Principles — Differentia  Hon. 


tangent.  At  the  points  for  which  the  tangent  is  parallel  to 
the  axis  of  x,  we  have  f  (x)  =  o ;  at  the  points  where  the 
tangent  is  perpendicular  to  the  axis,  f  (x)  =  co .  For  all 
other  points  f  (x)  has  a  determinate  finite  real  value  in 
general.  This  conclusion  verifies  the  statement,  that  the 
ratio  of  the  increment  of  the  dependent  variable  to  that  of 
the  independent  variable  has,  in  general,  a  finite  determinate 
magnitude,  when  the  increment  becomes  infinitely  small. 

This  has  been  so  admirably  expressed,  and  its  con- 
nexion with  the  fundamental  principles  of  the  Differential 
Calculus  so  well  explained,  by  M.  Navier,  that  I  cannot  for- 
bear introducing  the  following  extract  from  his  "Lecons 
d'Analyse": — 

"Among  the  properties  which  the  function  y  =  fix),  or 
the  line  which  represents  it,  possesses,  the  most  remarkable — 
in  fact  that  which  is  the  principal  object  of  the  Differential 
Calculus,  and  which  is  constantly  introduced  in  all  practical 
applications  of  the  Calculus — is  the 
degree  of  rapidity  with  which  the 
function  /  (x)  varies  when  the  in- 
dependent variable  x  is  made  to 
vary  from  any  assigned  value. 
This  degree  of  rapidity  of  the 
increment  of  the  function,  when  x 
is  altered,  may  differ,  not  only 
from  one  function  to  another,  but 
also  in  the  same  function,  ac- 
cording to  the  value  attributed  to 
the  variable.     In  order  to  form  a  s'    ' 

precise  notion  on  this  point,  let  us  attribute  to  a?  a  deter- 
mined value  represented  by  ON,  to  which  will  correspond 
an  equally  determined  value  of  y,  represented  by  PN.  Let 
us  now  suppose,  starting  from  this  value,  that  x  increases  by 
any  quantity  denoted  by  Ax,  and  represented  by  NM,  the 
function  y  will  vary  in  consequence  by  a  certain  quantity, 
denoted  by  Ay,  and  we  shall  have 

y  +  Ay  =  f(x  +  Ax),        or    Ay  =  f(x  +  Ax)  -fix). 

The  new  value  of  y  is  represented  in  the  figure  by  QM, 
and    QL  represents   Ay,  or  the  variation  of  the  function. 


Geometrical  Illustration.  g 

The  ratio  —  of  the  increment  of  the  function  to  that  of 

Ax 

the  independent  variable,  of  which  the  expression  is 

f(x  +  Ax)  -fix) 
Ax  ' 

is  represented  by  the  trigonometrical  tangent  of  the  angle 

QPL  made  by  the  secant  PQ  with  the  axis  of  x. 

Ay 
"  It  is  plain  that  this  ratio  —  is  the  natural  expression 

of  the  property  referred  to,  that  is,  of  the  degree  of  rapidity 
with  which  the  function  y  increases  when  we  increase  the 
independent  variable  x ;  for  the  greater  the  value  of  this 
ratio,  the  greater  will  be  the  increment  Ay  when  x  is  in- 
creased by  a  given  quantity  Ax.     But  it  is  very  important 

Ay 
to  remark,  that  the  value  of  —  (except  in  the  case  when 

Ax 

the  line  PQ  becomes  a  right  line)  depends  not  only  on  the 

value  attributed  to  x,  that  is  to  say,  on  the  position  of  P  on 

the  curve,  but  also  on  the  absolute  value  of  the  increment  Ax. 

If  we  were  to  leave  this  increment  arbitrary,  it  would  be 

Axt 
impossible  to  assign  to  the  ratio  -~  any  precise  value,  and 

it  is  accordingly  necessary  to  adopt  a  convention  which  shall 

remove  all  uncertainty  in  this  respect. 

"  Suppose  that  after  having  given  to  Ax  any  value,  to 

which  will  correspond   a   certain  value  Ay   and  a  certain 

direction  of  the  secant  PQ,  we  diminish  progressively  the 

value   of   Ax,    so   that  the    increment    ends  by  becoming 

evanescent ;    the  corresponding  increment  Ay  will  vary  in 

consequence,  and  will  equally  tend  to  become  evanescent. 

The  point  Q  will  tend  to  coincide  with  the  point  P,  and  the 

secant  PQ  with  the  tangent  PT  drawn  to  the  curve  at  the 

Ay 
point  P.      The  ratio  —    of   the    increments  will  equally 

LXX 

approach  to  a  certain  limit,  represented  by  the  trigonometrical 
tangent  of  the  angle  TPL  made  by  the  tangent  with  the 
axis  of  x. 

"We  accordingly  observe  that  when  the  increment  Ax, 


io  First  Principles — Differentiation. 

and  consequently  Ay,   diminish  progressively  and  tend  to 

vanish,   the  ratio  —    of  these    increments    approaches    in 

Ax 

general  to  a  limit  whose  value  is  finite  and  determinate. 

A7?/ 
Hence  the  value  of  —  corresponding  to  this  limit  must  be 

t\X 

considered  as  giving  the  true  and  precise  measure  of  the 
rapidity  with  which  the  function  f  (x)  varies  when  the  independent 
variable  x  is  made  to  vary  from  an  assigned  value ;  for  there 
does  not  remain  anything  arbitrary  in  the  expression  of  this 
value,  as  it  no  longer  depends  on  the  absolute  values  of  the 
increments  Ax  and  Ay,  nor  on  the  figure  of  the  curve  at  any 
finite  distance  at  either  side  of  the  point  P.  It  depends 
solely  on  the  direction  of  the  curve  at  this  point,  that  is,  on 
the  inclination  of  the  tangent  to  the  axis  of  x.  The  ratio 
just  determined  expresses  what  Newton  called  the  fluxion  of 
the  ordinate.  As  to  the  mode  of  finding  its  value  in  each 
particular    case,    it   is   sufficient  to    consider    the    general 

expression  .,        A   x      Ml  . 

r  Ay     f(x  +  Ax)  -j  (x) 

Ax  Ax 

and  to  see  what  is  the  limit  to  which  this  expression  tends, 
as  Ax  takes  smaller  and  smaller  values  and  tends  to  vanish. 
This  limit  will  be  a  certain  function  of  the  independent 
variable  x,  whose  form  depends  on  that  of  the  given  function 

f{x) We  shall  add  one  other  remark;  which  is,  that 

the  differentials  represented  by  dx  and  cly  denote  always 
quantities  of  the  same  nature  as  those  denoted  by  the  variables 
x  and  y.  Thus  in  geometry,  when  x  represents  a  line,  an 
area,  or  a  volume,  the  differential  dx  also  represents  a  line,  an 
area,  or  a  volume.  These  differentials  are  always  supposed 
to  be  less  than  any  assigned  magnitude,  however  small ;  but 
this  hypothesis  does  not  alter  the  nature  of  these  quantities  : 
dx  and  dy  are  always  homogeneous  with  x  and  y,  that  is  to 
say,  present  always  the  same  number  of  dimensions  of  the  unit 
by  means  of  which  the  values  of  these  variables  are  expressed." 
10a.  I<imit  of  a  Variable  Magnitude. — As  the  con- 
ception of  a  limit  is  fundamental  in  the  Calculus,  it  may 
be  well  to  add  a  few  remarks  in  further  elucidation  of  its 
meaning : — 


Limit  of  a  Variable  Magnitude.  1 1 

In  general,  when  a  variable  magnitude  tends  continually  to 
equality  with  a  certain  fixed  magnitude,  and  approaches  nearer  to 
it  than  any  assignable  difference,  however  small,  this  fixed  magni- 
tude is  called  the  limit  of  the  variable  magnitude.  For  example, 
if  we  inscribe,  or  circumscribe,  a  polygon  to  any  closed  curve, 
and  afterwards  conceive  each  side  indefinitely  diminished, 
and  consequently  their  rfumber  indefinitely  increased,  then 
the  closed  curve  is  said  to  be  the  limit  of  either  polygon. 
By  this  means  the  total  length  of  the  curve  is  the  limit  of 
the  perimeter  either  of  the  inscribed  or  circumscribed  polygon. 
In  like  manner,  the  area  of  the  curve  is  the  limit  to  the 
area  of  either  polygon.  For  instance,  since  the  area  of  any 
polygon  circumscribed  to  a  circle  is  obviously  equal  to  the 
rectangle  under  the  radius  of  the  circle  and  the  semi-perimeter 
of  the  polygon,  it  follows  that  the  area  of  a  circle  is  repre- 
sented by  the  product  of  its  radius  and  its  semi-circumfe- 
rence. Again,  since  the  length  of  the  side  of  a  regular 
polygon  inscribed  in  a  circle  bears  to  that  of  the  correspond- 
ing arc  the  same  ratio  as  the  perimeter  of  the  polygon  to  the 
circumference  of  the  circle,  it  follows  that  the  ultimate  ratio 
of  the  chord  to  the  arc  is  one  of  equality,  as  shown  in  Art.  9. 
The  like  result  follows  immediately  for  any  curve. 

The  following  principles  concerning  limits  *are  of  fre- 
quent application: — (1)  The  limit  of  the  product  of  two  quan- 
tities, which  vary  together,  is  the  product  of  their  limits;  (2)  The 
limit  of  the  quotient  of  the  quantities  is  the  quotient  of  their 
limits. 

For,  let  P  and  Q  represent  the  two  quantities,  and  p  and 
q  their  respective  limits ;  then  if 

P  =p  +  a,         Q  =  q  +  |3, 

a  and  j3  denote  quantities  which  diminish  indefinitely  as  P 
and  Q  approach  their  limits,  and  which  become  evanescent 
in  the  limit. 

Again,  we  have 

PQ  =  pq  +  pfi  +  qa  +  aj3. 

Accordingly,  in  the  limit,  we  have 

PQ=pq. 


12  First  Principles — Differentiation. 

Agam'  Q  =  q-Tp=q  +  a-(^)' 

The  numerator  of  the  last  fraction  becomes  evanescent  in 
the  limit,  while  the  denominator  becomes  q2,  and  consequently 

the  limit  of  —  is  -. 
Q     q 

1 1 .  Differentiation.— The  process  of  finding  the  derived 
function,  or  the  differential  coefficient  of  any  expression,  is 
called  differentiating  the  expression. 

We  proceed  to  explain  this  process  by  applying  it  to  a 
few  elementary  examples. 


Examples. 

i.  y  =  x2. 

Substitute  x  +  h  for  x,  and  denote  the  new  value  of  y  by  y\,  then 

y*L  =  (%  +  h)2  =  x2  +  2x7i  -f  h2  ; 

.    Vi-y      Ay 
.'.  — — -  or  — -  =  2x  +  h. 
h  Ax 

If  h  be  taken  an  infinitely  small  quantity,  -we  get  in  the  limit 

dy 

dx 

or  if  f(x)  =  x2,  we  have/'  (x)  =  zx. 

i 
2.  y=-. 

x 

_  i 

Here  y\  = 


x  +  h 


yi-y  = 


+  k      x  x  (x  +  7i) ' 


^  "  ^  or  ^  =  - 


h  Ax         x  (x  +  h) 

which  equation,  when  h  is  evanescent,  becomes 

dy  i  \xj  _        i 

dx  x2'  dx  x2 


Differentiation  of  a  Product.  13 

12.  Differentiation  of  tbe  Algebraic  Sum  of  a 
Finite  Number  of  Functions. — Let 

y=u+v-w+  &o. ; 
then,  if  xx  =  x  +  h,  we  get 

yx  =  Ul  +  Vi  -  w±  +  .  .  . ; 
V\—y      Ui-U      Vi—V      wl  —  w 

•'*  ~T~  =  ~T~  +  ~I  IT  +  '  '  *' 

which  becomes  in  the  limit,  when  h  is  infinitely  small, 

dy      du      dv      dw 
dx      dx      dx      dx 

Hence,  if  a  function  consist  of  several  terms,  its  derived 
function  is  the  sum  of  the  derived  functions  of  its  several  parts, 
taken  with  their  proper  signs. 

It  is  evident  that  the  differential  of  a  constant  is  zero. 

13.  Differentiation  of  tbe  Product  of  Vivo  Func- 
tions.— Let  y  -  uv,  where  u,  v,  are  both  functions  of  x ;  and 
suppose  Ay,  Au,  Av,  to  be  the  increments  of  y,  u,  v,  corre- 
sponding to  the  increment  Ax  in  x.     Then 

Ay  =  (u  +  Au)  (v  +  At?)  -  uv 

=  uAv  +  vAu  +  Au  Av, 

Ay        Av      ,  .  Au 

or  —  -u  —  +  (v  +  Av)  — . 

Afl?         Aa?  Ax 

Now  suppose  Ax  to  be  infinitely  small,  then 

Ay     Av     Au 

Ax'     Ax*     Ax* 
become  in  the  limit 

dy     dv        _  du 

ax     ax  ax 

also,  since  Av  vanishes  at  the  same  time,  the  last  term  dis- 
appears from  the  equation,  and  thus  we  arrive  at  the  result 

dy        dv        du 

dx       dx       dx'  ^' 


14  First  Principles — Differentiation. 

Hence,  to  differentiate  the  product  of  two  functions, 
multiply  each  of  the  factors  by  the  differential  coefficient  of  the 
other,  and  add  the  products  thus  found. 

Otherwise  thus :  let /(a?),  0  (a?),  denote  the  functions,  and 
h  the  increment  of  os,  then 

t/i  =  f(x  +  h)  <J>(x  +  h); 

.    Vi-y  =/(s  +  h)  (j>(%  +  h)  -f(x)  0  (a?) 
"       h  h 

Now,  in  the  limit, 


A 


=  /(*)>      <p(x  +  h)  =  0(a?), 


and                         0  (a?  +  A)  -  A  (a?)         ,, 
jj" =  0  Wi 

and,  accordingly, 

*-/(•)♦'(•)+♦(•)/«, 

which  agrees  with  the  preceding  result. 

When  y  =  au,  where  a  is  a  constant  with  respect  to  a?, 
we  have  evidently 

dy         du 

dx         dx' 

14.  Differentiation  of  the  Product  of  any  Number 
of  Functions. — First  let 

y  =  uvw; 

suppose  vw  =  z, 

then  y  =  uz, 

and,  by  Art.  13,  we  have 

dy         dz        du 
dx         dx        dx 9 


Differentiation  of  a  Quotient.  1 5 


but,  by  the  same  Article, 


dz  dv        dw 

hence 


dx  dx        dx 


dv  du  dv  dw 

JL   =   vw  —  +  WU  —  +  UV  — . 

dx  dx  ax  dx 

This  process  of  reasoning  can  be  easily  extended  to  any 
number  of  functions. 

The  preceding  result  admits  of  being  written  in  the  form 

1  dy       1  du      1  dv      1  dw 

y  dx       u  dx      v  dx     iv  dx9 

and  in  general,  if     y  =  yx .  y% .  y%  \  .  .  .  yn, 
it  can  be  easily  proved  in  like  manner  that 

1  dy       1  dyx      1  dy%  1  dyn  ,  * 

y  dx       y-L  dx      y2  dx  '  yn  dx' 

15.  Differentiation  of  a  Ctuotient — Let 

y  =  -,  then  u  =  yv ; 

„       „        ,      ,    ,  du         dv        dy 

therefore,  by  Art.  ,  3,    j-.g^+v^ 

dy 
or  v~ 

dx 


du        dv       du 

udv 

dx     ^  dx       dx 

vdx 

du        dv 

OjX            tvX 

V 

du        dv 

dy         dx        die 

(5) 


dx  v% 


This  may  be  written  in  the  following  form,  which  is  often 
useful : 

d  fu\      1  du      u  dv 

dx\vj     v  dx      v%  dx" 


1 6  First  Principles — Differentiation. 

Hence,  to  differentiate  a  fraction,  multiply  the  denominator 
into  the  derived  function  of  the  numerator •,  and  the  numemtor  into 
the  derived  function  of  the  denominator  ;  take  the  latter  product 
from  the  former,  and  divide  by  the  square  of  the  denominator. 

In  the  particular  case  where  u  is  a  constant  with  respect 
to  x  {a  suppose),  we  obviously  have 


d  fa\          a  dv 

(6) 

dx  \v)          v*  dx 

Examples. 

I# 

a  —  x 

u  =  . 

a  +  x 

A        du               2a 

ax          (a  +  xy 

2. 

u  —  (a  +  x)  (b-\-x). 

du             , 

— -  =  a  +  o  +  2X. 

dx 

16.  Differentiation   of  an   Integral   Power. — Let 

y  =  xn,  where  n  is  a  positive  integer. 

Suppose  yx  to  be  the  value  of  y,  when  x  becomes  xx,  then 

yiZJL   =    Xx     ~X      =  xn-i  +  xxn~i  +    .  .  .  +  flf-i. 

Now,  suppose  xx-  x  to  be  evanescent.  In  this  case  we 
may  write  x  for  xx  in  the  right-hand  side  of  the  preceding 
equation,  when  it  becomes  nxll~x\  but  the  left-hand  side,  in 

the  limit,  is  represented  by  -j-  • 

Hence  -j-  =  nx*1"1, 

ax 

d  (x*1) 
or  — V-1  =  nxn~l. 

ax 

This  result  follows  also  from  Art.  14  ;  for,  making 
Vx  =  2/2  =  Vz  =  •  •  •  =  Vn  =  u, 
we  evidently  get  from  (4), 


d  (un)  m  ,  du 


=  nu" 


dx  dx 

This  reduces  to  the  preceding  on  making  u  =  x. 


(7) 


Differentiation  of  a  Function  of  a  Function,  1 7 

17.  Differentiation  of  a  Fractional   Power. — Let 


y  =  un, 

then                   yn 

=  «*,  and  -^-z  =  -3— 7 ; 

hence,  by  (7), 

„  .  dy           m  .  du 
nyn~l-f  =  mu™"1  —  ; 
dx                dx 

d(un) 
dx 

dy       mum~xdu       m  --idu 
dx       n  yn"1  dx       n        dx 

(8) 

18.  Differentiation   of  a  Negative    Power. — Let 

y  =  er"*,  then  y  =  — ,  and  by  (6)  we  get 

U 

m  ,du 
mu™-1  — 
d  .  dx  m    du  ,  s 

s<*-) 5= «"r~s.         (9) 

Combining  the  results  established  in  (7),  (8),  and  (9),  we 
find  that 

\   7  =  mum-1  — 
dx  dx 

for  all  values  of  ^,  positive,  negative,  or  fractional.  When 
applied  to  the  differentiation  of  any  power  of  x  we  get  the 
following  rule : — Diminish  the  index  by  unity,  and  multiply  the 
power  of  x  thus  obtained  by  the  original  index  ;  the  result  is  the 
required  differential  coefficient,  with  respect  to  x. 

19.  Differentiation  of  a  Function  of  a  Function. — 

du 
Let  y  =  f(x)  and  u  =  0  (y),  to  find  — .    Suppose  yi,  ux,  to  be 

the  values  of  y  and  u  corresponding  to  the  value  xx  for  x ; 
then  if  Ay,  Au,  Ax,  denote  the  corresponding  increments, 
we  have  evidently 

Ui  -  u      ux  -  u  yt  -  y 


or 


xx-  x      y\  -  y  Xi  -  x9 


Au       Au  Ay 

Ax       Ay  Ax' 
o 


1 8  First  Principles — Differentiation. 

As  this  relation  holds  for  all  corresponding  increments, 
however  small,  it  must  hold  in  the  limit,*  when  Ax  is 
evanescent ;  in  which  case  it  becomes 

du       dudy  ,     N 

—  = -.  (io) 

dx       dy  dx 

Hence  the  derived  function  with  respect  to  x  of  u  is  the 
product  of  its  derived  with  respect  to  y ;  and  the  derived  of  y 
with  respect  to  x. 

20.  ^Differentiation   of  an  Inverse  Function. — To 

prove  that 

dx        1 

dy       d£ 

dx 

Suppose  that  from  the  equation 

y  =  f(x)  (a) 

the  equation 

x  =  $(y)  (b) 

is  deduced,  and  let  xl9  yl9  be  corresponding  values  of  x,  y, 
which  satisfy  the  equation  (a),  it  is  evident  that  they  will 
also  satisfy  the  equation  (b).    But 

V\-y     xl-  x  ___ 

%i-x    yx-y 

As  this  equation  holds  for  all  finite  increments,  it  must 
hold  when  xx-  x  and  yi-y  are  infinitely  small ;  therefore 
we  have  in  the  limit 

dy  dx  ,     ^ 

—  •—  =  1.  (11) 
dx  dy 

The  same  result  may  also  be  arrived  at  from  Art.  19, 
as  follows : — 

"When  y  =  f(x),  and  u  =  <j>(y), 


*  The  Student  -will  observe  that  this  is.  a  case  of  the  principle  (Art.  10a)  that 
the  limit  of  the  product  of  two  quantities  is  equal  to  the  product  of  their  limits. 


Differentiation  of  sin  x.  19 

we  have,  in  all  cases, 

du       du  dy 
dx       dy  dx 

This  result  must  still  hold  in  the  particular  case  when  u  =  x, 
in  which  case  it  becomes 

dxdy 

dy  dx 

Examples. 
1.  u  =  (a?  -  x2)5. 

Let  a2  —  x1  —  y,    then  u  —  y5, 


du           ,        ,  dy 

—  =  svS   and  —  =  -  2X. 

dy                      dx 

Hence 

—  =  -  iox  (a2  —  a;2)4. 

2. 

u   =  {a  +  bx*)*. 

3- 

u  —   (r  +  %2)i. 

4- 

u  =   (1  +  a")'". 

^4«5.   —  =  r25a;2  (a  +  bx2)*. 
dx 

du  x 


dx        (1  +  x2)^ 

du  , .  .     . 

—  =  mnxn-l(i+xn)m-1. 
dx 

We  next  proceed  to  determine  the  derived  functions  of 
the  elementary  trigonometrical  and  circular  functions. 
21.  Differentiation  of  sin  x. — Let 

y  =  sin#,      yx  =  sin  (x  +  h), 

.    h        (       h 

.    ,       7X       .  2sin-  cos   x+  - 

yx-  y  _  sin  (#  +  #)-  sin  x  _  2       \      2 

h  h  h 

.    h 
sm- 

But  by  Art.  9,  the  limit  of  — t—  =  1 ;  moreover,  the  limit  of 


X  +  -  J  is  cos  #. 

C  2 


20  First  Principles — Differentiation, 

Hence  a(sinx) 


=  cos  a?.  (12) 

dx 


22.  Differentiation  of  cos  x. 

y  =  cos  x,     yx  =  cos  (x  +  h), 


.    h    .    (       h^ 


f       ,  x  2  sin  -  sin  [x  +  - 

Vi  —  y  _  cos  (x  +  h)  -  cos  x  2        \       2  J 

h  h  h 

Hence,  in  the  limit, 

d  cos  x  ,    x 

-1-=-w,K.  (13) 

This  result  might  be  deduced  from  the  preceding,  by  substi- 

tuting %  f or  x3  and  applying  the  principle  of  Art.  19. 

It  may  be  noted  that  (12)  and  (13)  admit  also  of  being 
written  in  the  following  symmetrical  form : — 

dsinx       .    (      ir 


dx     =sm^+i'' 

d  cos  x  (      7r' 

=  cos  [x  +  -  1. 


dx  \       2 

23.  Differentiation  of  tan  x. 

y  =  tan  x,    y\  =  tan  (x  +  h), 

sin  (x  +  h)      sin  x 
yi-y     tan  (x  +  h)  -  tan  x  _  cos  (x  +  h)      cos  x 
~~h  h  h 

sin  h 


hco&x  cos  (x  +  h)9 


which  becomes  — =-  *&  the  limit. 
cos2# 


Differentiation  of  y  -  sin"1  x.  21 

-r-r  d(tana?)         1  ■  .     . 

Hence  — ~ — -  =  — z-  =  sec2#.  (14) 

dx  gq&x  v     ' 

Otherwise  thus, 

,  sin  #  d  sin  a?       .       6?  cos  x 

j,,         x       a . COS  X ; — -  -  sm  X ; — 

#(tan#)  cos  a?  ax     ■•  .  ax 


dx  dx  cos2  a? 


cos2  x  +  sin2  x 


cos2  x  cos2  x 


24.  Differentiation  of  cot  x. — Proceed  as  in  the  last, 

,  ,      d  (cot  x)  1  2  ,    A 

and  we  get     — y— — -  =  — r-r-  =  -  cosec2  #.  (15) 

dx  snrx 

This  result  can  also  be  derived  from  the  preceding,  by  put- 
ting —  z  f or  x,  as  in  Art.  22. 

25.  Differentiation  of  sec  x. 

1 

y  =  sec  x  = ; 

COS  X 

dy      max      .  .  ,. 

.*.  -r  =  — r—  =  tan  x  sec  a?.  (10) 

dx     cos2  x 

a.    .-,    1            d  cosec  x  . 

Similarly  ; =  -  cot  x  cosec  x. 

J  dx 

26.  Differentiation  of  y  =  snr1^. 

Here  x  =  sin  y,  .*.  -7-  =  cos  y. 

dy 

Hence,  by  Art.  20,  we  get 

dy  _      1  1 

dx     cos  y     ~  «/l  -x* 


22  First  Principles. — Differentiation. 

The  ambiguity  of  the  sign  in  this  case  arises  from  the  ambi- 
guity of  the  expression  y  =  sin"1  x ;  for  if  y  satisfy  this  equa- 
tion for  a  particular  value  of  x,  so  also  does  it  -  y;  as  also 
27r  +  y,  &c.  If,  however,  we  assign  always  to  y  its  least  value, 
i.  e.  the  acute  angle  whose  sine  is  represented  by  x,  then  the 
sign  of  the  differential  coefficient  is  determinate,  and  is  evi- 
dently positive  ;  since  an  angle  increases  with  its  sine,  so  long 
as  it  is  acute.    Accordingly,  with  the  preceding  limitation, 


d .  sin-1  x 


dx  y/j   _  #3 

In  like  manner  we  find 

d .  cos*"1  x  i 


(17) 


dx  y'j  _  x* 


(18) 


with  the  same  limitation. 

This  latter  result  can  be  at  once  deduced  from  the  preced- 
ing by  aid  of  the  elementary  equation 


sin  l  x  +  cos-10  =  -. 
2 


hence 


27.  Differentiation  of  tan-1  x. 

y  =  tan-1  x,  .*.  x  =  tan  y ; 
dx         1 


i  n. » 


Similarly, 


dy      cos2  y 

d .  tan'1  x     du  .  1  ,    x 

=  -r  =  cosiy=  x.  (19) 

dx  dx  1  +  x%  x     ' 

e? .  cot-1  a?  1 

db  1  +  x2' 


28.  Geometrical   Demonstration. — The   results  ar- 
rived at  in  the  preceding  Articles  admit  also  of  easy  demon- 


Geometrical  Demonstration. 


23 


stration  by  geometrical  construction.     We  shall  illustrate  this 
method  by  applying  it  to 
the  case  of  sin  0. 

Suppose  XP  QY  tohe  a 
quadrant  of  a  circle  hav- 
ing 0  as  its  centre,  *and 
construct  as  in  figure. 
Let  0  denote  the  angle 
XOP  expressed  in  circu- 
lar measure ;  then  Fig.  2. 

„     arc  PX       ,  ,        ~     arc  PQ 
0  =  — tt^t- ,  and  h  =  At/  = 


OP 


OP 


Accordingly, 
sin  (0  +  h)  -  sin  0  =  -^  =  ^  .  -^  =  cos  P  QR .  ^ ; 

.    sin(0  +  A)-sinfl  =  cog pQR     m 
h  arc  PQ 

PQ 

But  we  have  seen,  in  Art.  9,  that  the  limiting  value  of ^-x 

arc  JrQ 

=  1  ;  also  PQR  =  0,  at  the  same  time  ;  hence  — ^ —  =  cos  0, 

as  before. 

The  student  will  find  no  difficulty  in  applying  the  pre- 
ceding construction  to  the  differentiation  of  cos  0,  sin-1  0,  and 
cos"1  0.  The  differential  coefficients  of  tan  0,  tan"1  0,  &c,  can, 
in  like  manner,  be  easily  obtained  by  geometrical  construction. 


1.  y  —  sin  (nx  +  a). 

2.  y  =  cos  mx  cos  nx. 

3.  y  =  sinw#. 


Examples. 

—  =  n  cos  ma?  +  a), 
ax 


dy 

dz 

dy 
dx 


=  -  {m  cos  nx  sin  m#  +  w  cos  mx  sin  ft#), 
=  n  sinM-1  a;  cos  #. 


24  First  Principles — Differentiation. 

4.  y  -  sin  (1  +  a2).  -^  =  23  cos  (1  +  #2). 

5.  Show  that  sin3  x  — -  (sinm#  sin  mx)  =  m  $mm+lx  sin  (w  +  j)  x. 

_       d  ,  . 

Here  —  (sm"*#  sm  mx)  —  m  sinw_1a;  (cos  x  sin  mx  +  sin  x  cos  m#) 

=  m  sin"8-1  x  sin  (m  +  1)  x :  .*.  &c. 

6.  «/  =  O  sin8  x  +  b  cos2  #)».  ~  =  n(a-b)  sin  22:  («  sin2  x  +  b  cos2  a;)"-1. 


7.  ?/  =  sin  (sin  #). 

=  sin#. 


Or  «/  =  sin  u,  where  w  =  sin  x.  —-  =  cos  x  cos  (sin  3). 

dx  v        ' 


8.  ar  =  sin-1  (#w). 

v    '  tf#      (1  -  a2")* 

9.  y  =  sin-1  (1  —  #2)*. 

Here  (1  -  #2)^  =  sin  y ;  .*.  .r  =  cos  y. 

dy                       dy  \ 

i=-smy-?-;  .-.—  = ■ 

dx  dx     yr^ 

.  b  +  a  cos  x  dy      .  /Tjj      I£ 

10.  y  =  cos-1  — — : .  ~-  =  v  g       6 

a  +  £cos#  <fe     ^+acosa;* 

<&/ 

11.  «/  =  seert  x.  -f-=n  secM#  tan x. 

dx 

12.  «/  =  sec-1  (a2).  ■— ■=       /— . 

9  dx      x^/x^-i 

29.  Differentiation  of  log0a?. 

Let  y  =  logax,    yx  =  log0  (a?  +  ^), 

Vi-y  =  loga  (a?  +  h)  -  logqX  =    °gg\    +  x)  m 
h  h  h 

Hence  -~  is  equal  to  the  limiting  value  of 
dx 


Jl0^(I+i) 


when  h  is  infinitely  small. 
Again,  let  h  =  xu,  then 


1 ,       /      h\      1  loga  (1  +  u)      1.       ,  ± 

-l0gj  I  +  -     =  -  -2-i '-  =  -  l0ga  (I  +  U)\ 

h         \      xj     x         u  x     °   x 


Geometrical  Demonstration.  25 

j  1 

.\  ~  =  -  multiplied  by  the  value  of  log«  (1  +  u)u  when  u  is 

(IX      x 

infinitely  small. 

To  find  the  value  of  the  latter  expression,  let  -  =  s,  then 

u 

(1  +  u)u  becomes  ( 1  +  -) ,  in  which  z  is  regarded  as  infinitely 

great.  Suppose  the  limiting  value  of  this  expression  to  be  re- 
presented  by  the  letter  e,  according  to  the  usual  notation.  We 
can  then  find  the  value  of  e  as  follows  by  the  Binomial 
Theorem : — 

i\s  %     1      z  (z  -  1)     1 

1  +-    =  1  +  -.-  +  — ^ '-  .-+... 

\        %)  I     Z  I  .  2         z2. 


+  &G. 

The  limiting*  value  of  which,  when  z  =  00,  is  evidently 

111  1  _ 

1  +  -  +  +  +  +  &c. 

I         1.2         1.2.3         I.2.3.4 

By  taking  a  sufficient  number  of  terms  of  this  series,  we 
can  approximate  to  the  value  of  e  as  nearly  as  we  please. 
The  ultimate  value  can  be  shown  to  be  an  incommensurable 
quantity,  and  is  the  base  of  the  natural  or  Napierian  system 
of  logarithms.  When  taken  to  nine  decimal  places,  its  value 
is  2.718281828. 


Again,  since  (1  +  u)u  =  e  when  u  =  o,  we  get 

d.loggx  ^  logae 

dx  x 


(20) 


Also,  since  the  calculation  of  logarithms  to  any  other 
base  starts  from  the  logarithms  of  some  numbers  to  the  base  e ; 

*  It  will  be  shown  in  Chapter  3,  without  assuming  the  Binomial  expansion, 
that  e  is  the  limit  of  the  sum  of  the  series 

11  1  „ 

1  -I 1 + V  &c,  ad  infinitum. 

1      1.2      1.2.3 


26  First  Principles — Differentiation. 

and  moreover,  since  the  logarithms  of  all  numbers  are  expressed 
by  their  logarithms  to  the  base  e  multiplied  by  the  modulus 
of  transformation,  the  system  whose  base  is  e  is  fundamental 
in  analysis,  and  we  shall  denote  it  by  the  symbol  log  without 
a  suffix.     In  this  ease,  since  log  e  =  i,  we  have 


Again, 


—  (Logos)  =  -.  (21) 

ax      °   '      x 


d  n         x     logl0e     If  .    . 

—  log10a?)  =  — 2—  =  — ,  (22) 

ax  x  x 


where  M  or  log10  e  is  the  modulus  of  Briggs'  or  the  ordinary 
tabulated  system  of  logarithms.  The  value  of  this  modulus, 
when  calculated  to  ten  decimal  places,  is 

0.4342944819. 

On  the  method  of  its  determination  see  Gralbraith's  "Algebra," 

P-  379* 

If  a?  be  a  large  number,  it  is  evident,  from  the  preceding, 

that  the  tabular  difference  (as  given  in  Logarithmic  Tables), 

M 
i.  e.  the  difference  between  logio  (#  +  1)  and  logi0#,  is  — ,  ap- 

x 

proximately.     The  student  can  readily  verify  this  result  by 

reference  to  the  Tables. 

30.  Differ entiation  of  ax. 

Let  y  =  ax,    then  log  y  =  x  log  a ; 

tut  ^(log  y)  =  d  (log  y)  dy  _  1  dy  % 

dx  dy      dx      y  dx' 

d  .ax      dy        ,  _ ,  t     x 

'"'  ~dx~  =tc  =  yga=         g  a'  ^23) 

Also,  since  log  e  =  1,  we  have 

d    & 


Logarithmic  Differentiation.  27 


Examples. 


I.  y  —  log  (sin#). 

Let  sin  x  =  z,  then  y  =  log  2. 

<&/      %    dz 
dx      dz  '  dx* 


And  since 


we  get  —  —  — —  =  cot  x. 

0  dx      sin  x 


dy      cos  x 

—  COL  X. 

dy 


2.  7/  =  log  \/«3  -  ^3  =  | log  (a2  -  #2) ; 


dx         a1  —  #2 


.       dy 
J    J  dx 


ll  —  cos* 
y  ~  og  \  1  +  cos  a; 


J 


2  sm"- 


aj 


COS  X  \  2  X 

tan  -; 


1  +  cos  x        /         „x  2 

^/  2  cos*5  - 


.*.  y  —  log  tan-.     Hence  —  =  — — • 
.2  dx      smx 

31.  logarithmic  differentiation. — When  the  func- 
tion to  be  differentiated  consists  of  products  and  quotients 
of  functions,  it  is  in  general  useful  to  take  the  logarithm 
of  the  function,  and  to  differentiate  it.  This  process  is  called 
logarithmic  differentiation. 

Examples. 

1.  y-yi-yz-n'-'l/n,        log  y  =  log  y\  +  log  y%  +  .  .  .  +  log  y„. 

Hence  I*  =  1  &  +  1*?  + .  .  .  +  -L  ^». 

This  furnishes  another  proof  of  formula  (4),  p.  15. 

2.  y  — .    Here,  log  y  —  m  log  sin  x  —  n  log  cos  x ; 

9      cosw  a;  ° 

I  $y  cos  a;         sin  x  dy      sin™-1 #  .         „  .  „    , 

.•.  -  —  =  m +  n ;  .*.  -f-  =  r-  (mcos2#  +  wsm^. 

y  dx         sin  x        cos  x         dx      cosmi  x 


28  First  Principles — Differentiation. 

(x  -  i)i 


3-  y- 


hence 


(x  -  2)1  {x  -  3)1* 


5  3  7 

Here  log  2/  =  -  log  (a;  -  1)  -  -  log  (x  -  2)  —  log  (a?  -  3) ; 

2  4  3 


1  dy      $      1  31  7i  7#2  +  3°#  -  97 


2/  $£      2«-i      4  a;  —  2       3  #  —  3         12  .  (#  —  1)  {x  —  2)  (x  -  3)  ' 

b    %  _      (a;  -  i)-§  (7a;2  +  30a;  -  97) 
'  '  dx  \2.{x-  2)1  (a;  -  3)^ 

dy      cfi  +  a?  x2  -  4a;4 

4.  y=x{a?  +  x*)</a?-x\  ^=  — ^==— . 

5.  ^  =  a;*.     Here  log  y  =  x  log  #. 

„  1  dy  .  d .  xx 

Hence  -  —  =  (log  x  +  1) ;         .*.  — —  =  xx  (1  +  log  a:). 


6.  «/  =  exX.    Here  log  y  =  xx, 

1  dy     d.xx 


y  dx       dx 


=  xx  (r  +  logo;); 


dy      *    /      1      x 

.*.  ~  =  ex  xx  (1  +  log  an. 
da? 

7.  y  =  uv,  where  ««  and  #  are  both  functions  of  x. 

Here  log  y  —  v  log  u, 

1  dy  dv      v  du 

y  dx  dx      u  dx* 

dy  /,         dv      v  du\  .         dv  .  du 


(dv      v  du\  . 

loguTx  +  ld~x)  =uVl° 
tv*v  iff  U/U/  / 


dx  \  dx      u  dx)  dx  dx 

32.  The  expression  to  be  differentiated  frequently  admits 
of  being  transformed  to  a  simpler  shape.  In  such  cases  the 
student  will  find  it  an  advantage  to  reduce  the  expression  to 
its  simplest  form  before  proceeding  to  its  differentiation. 

Examples. 

v. 
1.  y  =  sur1 


\/i  +  x2' 


Here  — ■===.  —  sin  «/,  or =  sin2  y  ;  hence  x  =  tan  y, 

dy       *  l 

and  we  get  —  =  cos2  y  = 


dx  1  +  a;2 


Hence 


Logarithmic  Differentiation.  29 


2.  y  —  tan-1 


Here  tan  y  = 


*f  1  +  x2  —  \/i  —  x1 

*fl  +  £C2  +  \/ 1  -  x% 
s/\  +  x2  -  *y  \  -  xz 

*y 1  4-  x2  _  tan  y  +  1  % 
i/T^x~*  ~  tan  y  -  i ' 

.      (1  +  tan  #)2  -  (1  -  tan«/)2        2  tan  2/ 

.•.  %*  =  -^ — —  = —  =  sin  2«/. 

(1  +  tan  yf  +  (i  -  tan  y)*      i  +  tan*?/ 

Hence  —  cos  2*/  =  x. 

dx 

dy  _      x  x 


dx     cos  2y      yx  _  xi 


\\/ I  +  X  +  \/ I  —X 

-log        y , 

\ y  i+x-^/i— x 


—  X      1  n      v^1  +  #  +  a/1  - 

=  7  loS 


-v/i  +  #  —  v  I  -  # 


I  I  +  */\  -  X*        I  .        ,  / 5.         I  . 

=  -  log - =  -  log  (1  +  y  1  -  x2)  —  log  x. 


dy  1 

dx  23  Vi  -  x2 


.  y/i  +  #2  -  i      ,      .     2# 

4.  #  =  tan"1 +  tan-1 -. 

x  1  —  xi 

Let  x  =  tan  z,  and  the  student  can  easily  prove  that 


5       ,          dy     s      1 
•■  -  z ;  hence  —  = s 


3<d  First  Principles — Differentiation. 


Examples. 


i.  y  =  sec-1  x. 


dx 


V'. 


x\/  x 


dy  . 

2.  y  —  x  log  x.  -—  =  I  +  log  x. 

dx 

3.  y  =  log  tan  x.  —  =  -7— - ■ . 

«#      sin  2x 


4.  &/  =  log  tan*1  a:. 

5.  y  =  a</x. 

6.  y  =  sin  (log  a). 

7.  */  =  tan"1 


#  1 


da; 

(1  +  #2)  tan"1  x 

da? 

a 
zyx 

d«/  _ 

dx 

cos  (log  x) 

1  • 

dy 

I 

dx 

\/l-X2 

dy 

I 

dx 

2/y/ a;  (1  +  %)* 

Y  1  —  x2 

8.  «/  =  tan-1 — —  • 

I  —  V  CM 

Here  ^  =  tan~~  V  x  +  *an_1  V^* 

x2*1  dy        2nx2n~1 

9'  y=  (1  +  z*y  dx  ~  (1  +  #2)«+-* 


10.  ^log^—j   -*taa-*«.    ^=--^. 

11.  y  =  log    /    x 

\V  1  +  a:3- 


j»  dy  _         1 

0  ^      </l+xi 


.3  +  2-;  % 

12.  2/  =  Sin1- rrr 


^13  *&      y^i  —30- 


a* 


13.  w  =  log  7 77-  +  A  tan-1  x.  —  =  7 r-7 ^r. 

J    9        5  (1  +  a;)i       2  da;      (1  +  a;)  (1  +  a;2) 

1  -a;  #_     (1  +  a?) 

I4'y"*/rT^'  d*_~(i +**)§* 


Examples.  3 1 


(c  -  x2)%  sin-1  x        .        dy      1  -  x-      1  +  2X-  .         ... 

1  c.  11=  - - •.     Ans.  —  =  • (i  -  x2)i  .  sin'1  x. 

J    J  x  dx  x  x°~  ' 

-  1  —  tan  x  dy  .  .      . 

16.  y  =  .  —  =  -  (cos  a;  +  sm«. 

sec  x  dx 

\/i-x2  +  x\/2      dy  \/ '2 

17.  y  =  log z==== .    —  = ■ — . 

*/i-x2  dx      Wi-x*  +  x*/2){i-xi) 

gatan     x  (#,%  _  j}  fly        (j  4.  a2)  x  ea  tan     * 

1    '  V  =  (l  +  X2)*        '  ~dx =  (I  +  *2)f  * 

,      i  +  x      ..      i  4-  x  +  x2         ,—        ,x\/z         dy         6 

19.  y  =  log +  \  log +  V3  tan-'  — ^—±.        -f-  = ■ 

0  1  -  x      *     °  1  -  x  +  x~  i-x2        dx      1  -x5 

dy  __  1 

da;      (x2  —  x  —  i)i' 

li  +  x  \/ 2  4-  a;2  .  x\/ 2 

21.  «/  = 


20.    2/  =  log  {(2£  -  i)  +  2\/ X2  -X  -  i}. 


/i  +  a;  y/ 2  +  x2  _xx\/z  dy  _2*y  2 

\  1  —  x  a/^2  +  #2  i  —  #2"  dx      1  +  x*' 

x  dy        x  /     1  \ 

22.  «/  =  ex  tan-'  a?.  -j~  =  &c    ( %  +  xx  tarrl  x  (J  +  loS  ^) )  • 

23.  Being  given  that  y  =  a;3  ( 1  -  x2  J 1 1  -  —  j    ;  if 

%         ex2  +  c'x*  +  c"x6 


dx       /         „\W        x2\i' 


('-f('-r) 

determine  the  values  of  c,  e',  c".    Ans.  c  =  3,  c'  =  -  6,  c"  =  f . 
24.  y'=log(log#). 


,  3  +  5  cos  a: 
25.  y  =  cos  * . 


dx     x  log  x* 
dy  4 


5  +  3  cos  x'  dx     5  +  3  cosa;* 

.    .  r  -  x*  dy        -  2 

26.  w  =  sir1 -.  — -  =  ■ . 

*  1  +  a2  <?#      1  +  a;2 

dy 

27.  y  =  eax  sin™  ra.  -f^-  =  e««  sin"1"1  ra  (a  sin  ra;  +  w  cos  ra?) . 

dx 

38.  «/  =  eaxsmr$.  —■  =  eax*s/a2  +  r*sm.  (rx  +  <J>), 

where  tan  <b  =  -. 


32  First  Principles — Differentiation, 

29.  y  =  log (-/*-•  +  V* - *).      Ans.  dx - 2 v/(g,fl)(g.^ 


30.  y  =  2  tan-1 


/i-x\l 

\i  +  ^y 


Here  =  tan2  - ;  .\  a;  =  cos  y ;  .*.  ^j-  —  - 


1  +  a:  2  *  <fcr  (i  -  x2)* 

31.  2/  =  %x  •  j~xX        (n  1°S  x  +  l)- 

CLX 

m  ,  m  - 1 

32.  y  —  (1  +  a;2)2 sin  (m  tan-1  #).        -f-  =  m(l  +  a;2)  2  cos{(m-i)tan-1ip}, 


,         /«  cos  x  —  b  sin  x  dy  —  ab 

l.y  =  loS^j- 


?•?.   y  =  log     / .  —  =  — — — — — — — — —  • 

*Sa  cos  a;  +  b  sin  x  dx      a2  cos2  x  —  b%  sin2  x 


34.  Define  the  differential  coefficient  of  a  function  of  a  variable  quantity, 
•with  respect  to  that  quantity,  and  show  that  it  measures  the  rate  of  inerease  of 
the  function  as  compared  with  the  rate  of  increase  of  the  variable. 


35 .  If  y  =  -,  prove  the  relation 
x 


dy  dx 

+  — 7=r  =  o. 


a/ l  +  y*>      ^/i  +  x* 


*      T*  1         X*  +  aX  +  \S(X2  +  aXY  ~bX  ,U    4.    dU       ■  V    A         t 

36.  If  u  =  log -1  prove  that  —    is  of  the   form 

x*  +  ax  -  ^{x2  +  ax)i  -  bx  dx 

Ax  4-  B 

,  and  determine  the  values  of  A  and  B.    Ans.  A  —  3,  B  =  a. 


\/ (x2  +  ax)2  —  bx 

d  f  . \      A  sin* 9  +  .Bsin20  +  C 

37.  Prove  that  -  ^sin  e  cos  e  Jx  _  #  sin2  0J  =  —^===^-  , 

and  determine  the  values  of  A,  B,  C.      Ans.  A  =  3c3,  B  =  -  2  (1  +  c2),  C  =  1. 

■?8.  I£u  =  x  + H — f-  — - — 7  —+...«<?  «^f.  ;  find  the  sum 

23      2.45      2.4.67' 

of  the  series  represented  by  — .  Ans.  (1  -  32)~*. 

39.  Eeduce  to  its  simplest  form  the  expression 

-la?  d     x(x2  +  2a)%  .  I 

Ans.  —       


(a;2  +  «)i  (x2  +  lafi      dx  '     (x2  +  a)l  '  '  (x2  +  a)i  (a;2  +  aa)i ' 

.  .    »     ,-•  ,  dy     sin2  (a  +  y) 
40.  If  sin  y  =  x  sin  {a  4-  2/),  prove  that  -—  = : . 

ClX  S1H  ^ 


Examples,  33 

41.  If  #(1  +  ^  +  ^(1  -\-x)i  =  o,  find-^. 

€L0(/ 

In  this  case  x2  (1  +  y)  =  y2  (1  +  x) ; 

.*.  x2  -  y2  =  yx  (y  —  x), 

or  x  +  y  +  xy  =  o ;     .*.  y  = ;   .  •.  —  =  —  ; . 

9        9  9  i+x  dx  (1  -i-  x2) 

1      /  /  "5 ^  t  x  ty      l     \x  +  a 

42.  y  =  log  (x  +  v  a-2  -  a2)  4-  sec"1  -.  —  =  -  a  • 

ox  '  a  dx      x^x-a 

43.  If  x  and  y  are  given  as  functions  of  t  by  the  equations 

x=f(t);    y  =  F{t); 

find  the  value  of  —  in  terms  of  t.  —  =  -tt-tt  • 

*?#  ##     /  (t) 

44-  y  =  ITP 


I  +  x* 


Hence 

2/  = 

X* 

i+y 

x-y 

45- 

X  - 

=  ey . 

Hence 

y- 

X 

l  +  &c,  ad  infinitum, 
dy  x 


dx      <\/x*  +  i 


dy  log  x 

dx~  (1  +  logs?)2* 


D 


(     34     ) 


CHAPTER  II. 

SUCCESSIVE   DIFFERENTIATION. 

33.  Successive  IBerived  Functions. — In  the  preceding 
chapter  we  have  considered  the  process  of  finding  the  derived 
functions  of  different  forms  of  functions  of  a  single  variable. 

If  the  primitive  function  be  represented  by /(a?),  then,  as 
already  stated,  its  first  derived  function  is  denoted  by  /'(#). 
If  this  new  function,  fix)-,  be  treated  in  the  same  manner, 
its  derived  function  is  called  the  second  derived  of  the  original 
function /(a?),  and  is  denoted  by /"(a?). 

In  like  manner  the  derived  function  of  f'{%)  is  the  third 
derived  of  /(#),  and  represented  by /'"(a?),  &c. 

In  accordance  with  this  notation,  the  successive  derived 
functions  of /(a?)  are  represented  by 

/»,   /»,   /'», — /«(•), 

each  of  which  is  the  derived  function  of  the  preceding. 
34.  Successive  Differential  Coefficients. 

If  y  =  /(aOwehave^=/». 

Hence,  differentiating  both  sides  with  regard  to  x9  we  get 

i. (f\ .  '/>(.)  =/». 

Let         l(l)berePresentedbyS' 

then  g  =/"(,)• 

In  like  manner  —  f  -^  ]  is  represented  by  — ,  and  so  on  ; 


Successive  Differentials.  35 

henoe     2 =r{x)' &o-  •  •  •  S =/(n)  w-     w 

The  expressions 

^     <^V     <#3^  dny 

da?     da?'     dxz<>  dxn 

are  called  the  firsts  second,  third,  .  .  .  nth  differential  coef- 
ficients of  y  regarded  as  a  function  of  x. 

These  functions  are  sometimes  represented  by 

•,  y",  y"',  . . .  y«, 

a  notation  which  will  often  be  found  convenient  in  abbre- 
viating the  labour  of  forming  the  successive  differential 
coefficients  of  a  given  expression.  From  the  mode  of 
arriving  at  them,  the  successive  differential  coefficients  of  a 
function  are  evidently  the  same  as  its  successive  derived 
functions  considered  in  the  preceding  Article. 

35.  Successive  Differentials. — The  preceding  result 
admits  of  being  considered  also  in  connexion  with  differen- 
tials ;  for,  since  x  is  the  independent  variable,  its  increment, 
dx,  may  be  always  taken  of  the  same  infinitely  small  value. 
Hence,  in  the  equation  dy  =  f(x)  dx  (Art.  7),  we  may 
regard  dx  as  constant,  and  we  shall  have,  on  proceeding 
to  the  next  differentiation, 

d  {dy)  =dxd  [/'  (a?)]  =  (dx)2  f"(x), 

since  d  [/'  (a?)]  =/"  (x)  dx. 

Again,  representing       d  (dy)  by  d2y, 

we  have  d2y  =  f" (x)  (dx) 2 ; 

if  we  differentiate  again,  we  get 

d*y=f"(x)(dx*); 
and  in  general 

dny=/W(x)(dx)n. 

Prom  this  point  of  view  we  see  the  reason  why/(n)  (x)  is 
called  the  nth  differential  coefficient  oif(x). 

d2 


36  Successive  Differentiation. 

In  the  preceding  results  it  may  be  observed  that  if  dx 
be  regarded  as  an  infinitely  small  quantity,  or  an  infinitesimal 
of  the  first  order,  {dx)2,  being  infinitely  small  in  comparison 
with  dx,  may  be  called  an  infinitely  small  quantity  or  an 
infinitesimal  of  the  second  order ;  as  also  d2y,  if  /"  (x)  be 
finite.  In  general,  dny,  being  of  the  same  order  as  (dx)n,  is 
called  an  infinitesimal  of  the  nth  order. 

36.  Infinitesimals. — We  may  premise  that  the  expres- 
sions great  and  small,  as  well  as  infinitely  great  and  infinitely 
small,  are  to  be  understood  as  relative  terms.  Thus,  a  magni- 
tude which  is  regarded  as  being  infinitely  great  in  comparison 
with  definite  magnitude  is  said  to  be  infinitely  great.  Similarly, 
a  magnitude  which  is  infinitely  small  in  comparison  with  a 
finite  magnitude  is  said  to  be  infinitely  small.  If  any  finite 
magnitude  be  conceived  to  be  divided  into  an  infinitely  great 
number  of  equal  parts,  each  part  will  be  infinitely  small  with 
regard  to  the  finite  magnitude ;  and  may  be  called  an  infini- 
tesimal of  the  first  order.  Again,  if  one  of  these  infinitesimals 
be  conceived  to  be  divided  into  an  infinite  number  of  equal 
parts,  each  of  these  parts  is  infinitely  small  in  comparison 
with  the  former  infinitesimal,  and  may  be  regarded  as  an 
infinitesimal  of  the  second  order,  and  so  on. 

Since,  in  general,  the  number  by  which  any  measurable 
quantity  is  represented  depends  upon  the  unit  with  which 
the  quantity  is  compared,  it  follows  that  a  finite  magnitude 
may  be  represented  by  a  very  great,  or  by  a  very  small  num- 
ber, according  to  the  unit  to  which  it  is  referred.  For  ex- 
ample, the  diameter  of  the  earth  is  very  great  in  comparison 
with  the  length  of  one  foot,  but  very  small  in  comparison 
with  the  distance  of  the  earth  from  the  nearest  fixed  star,  and 
it  would,  accordingly,  be  represented  by  a  very  large,  or  a 
very  small  number,  according  to  which  of  these  distances  is 
assumed  as  the  unit  of  comparison.  Again,  with  respect  to 
the  latter  distance  taken  as  the  unit,  the  diameter  of  the 
earth  may  be  regarded  as  a  very  small  magnitude  of  the  first 
order,  and  the  length  of  a  foot  as  one  of  a  higher  order  of 
smallness  in  comparison.  Similar  remarks  apply  to  other 
magnitudes. 

Again,  in  the  comparison  of  numbers,  if  the  fraction  (one 

million)^  or  — -6,  which  is  very  small  in  comparison  with 


Geometrical  Illustration. 


37 


unity,  be  regarded  as  a  small  quantity  of  the  first  order,  the 

fraction  — -,  being  the  same  fractional  part  of  — 6  that  this 

is  of  i,  must  be  regarded  as  a  small  quantity  of  the  second 
order,  and  so  on. 

'-Y 


If  now,  instead  of  the  series  — -,  (  — -  ) , 

io6    Vio6/ 


we    consider    the    series   -,  — ,  — , 

n    w    w 


\IQr, 

in  which   n   is 


supposed  to  be  increased  without  limit,  then  each  term  in  the 
series  is  infinitely  small  in  comparison  with  the  preceding 
one,  being  derived  from  it  by  multiplying  by  the  infinitely 

small  quantity  -.  Hence,  if  -  be  regarded  as  an  infinitesimal 

of  the  first  order,  — „,  —,...—-,  may  be  regarded  as  infini- 

nr  nz  nr 

tesimals  of  the  second,  third,  .  .  .  rth  orders. 

37.  Geometrical  Illustration  of  Infinitesimals. — 

The  following  geometrical  results  will  help  to  illustrate  the 
theory  of  infinitesimals,  and  also 
will  be  found  of  importance  in  the 
application  of  the  Differential  Cal- 
culus to  the  theory  of  curves. 

Suppose  two  points,  A,  B,  taken 
on  the  circumference  of  a  circle ; 
join  B  to  E9  the  other  extremity 
of  the  diameter  AE,  and  produce 
EB  to  meet  the  tangent  at  A 
in  D.  Then  since  the  triangles 
ABB  and  EAB  are  equiangular, 
we  have  Fig.  3. 

AB     BE  BD      AB 

AD  ~  AE'         AB  ~  AE' 

Now  suppose  the  point  B  to  approach  the  point  A  and  to 
become  indefinitely  near  to  it,  then  BE  becomes  ultimately 

AB 


equal  to  AE,  and,  therefore,  at  the  same  time, 


AD 


=  1. 


38  Successive  Differentiation. 

Again,  -j=r  becomes  infinitely  small  along  with  -j=, 

i.  e.  BD  becomes  infinitely  small  in  comparison  with  AD  or 
AB.  Hence  BD  is  an  infinitesimal  of  the  second  order  when 
AB  is  taken  as  one  of  the  first  order. 

Moreover,  since  DE  -  AE  <  BD,  it  follows  that,  when  one 
side  of  a  right-angled  triangle  is  regarded  as  an  infinitely  small 
quantity  of  the  first  order,  the  difference  between  the  hypothenuse 
and  the  remaining  side  is  an  infinitely  small  quantity  of  the 
second  order. 

Next,  draw  BN  perpendicular  to  AD,  and  BF  a  tan- 
gent at  B;  then,  since  AB  >  AN",  we  get  AD  -  AB 
<AD-AN<DN; 

AD-  AB     DJST     AD 
BD       <  BD<  DE' 

Consequently, — —  becomes  infinitely  small  along  with 

AD;  .'.  AD  -  AB  is  an  infinitesimal  of  the  third  order. 
Moreover,  as  BF  =  FD,  we  have  AD  =  AF  +  BF;  .*.  AF 
+  BF  -  AB  is  an  infinitely  small  quantity  of  the  third  order  ; 
but  AF  +  FB  is  >  arc  AB,  hence  we  infer  that  the  difference 
between  the  length  of  the  arc  AB  and  its  chord  is  an  infinitely 
small  quantity  of  the  third  order,  when  the  arc  is  an  infinitely 
small  quantity  of  the  first.  In  like  manner  it  can  be  seen 
that  BD  -  BN  is  an  infinitesimal  of  the  fourth  order,  and 
so  on. 

Again,  if  AB  represent  an  elementary  portion  of  any 
continuous*  curve,  to  which  AF  and  BF  are  tangents,  since 
the  length  of  the  arc  AB  is  less  than  the  sum  of  the  tangents 
AF  and  BF,  we  may  extend  the  result  just  arrived  at  to  all 
such  curves. 

*  In  this  extension  of  the  foregoing  proof  it  is  assumed  that  the  ultimate 
ratio  of  the  tangents  drawn  to  a  continuous  curve  at  two  indefinitely  near 
points  is,  in  general,  a  ratio  of  equality.  This  is  easily  shown  in  the  case  of 
an  ellipse,  since  the  ratio  of  the  tangents  is  the  same  as  that  of  the  parallel 
diameters.  Again,  it  can  be  seen  without  difficulty  that  an  indefinite  number 
of  ellipses  can  be  drawn  touching  a  curve  at  two  points  arbitrarily  assumed  on 
the  curve  ;  if  now  we  suppose  the  points  to  approach  one  another  indefinitely 
along  the  curve,  the  property  in  question  follows  immediately  for  any  con- 
tinuous curve. 


Geometrical  Illustration.  39 

Hence,  the  difference  between  the  length  of  an  infinitely 
small  portion  of  any  continuous  curve  and  its  chord  is  an  infi- 
nitely small  quantity  of  the  third  order,  i.e.  the  difference  between 
them  is  ultimately  an  infinitely  small  quantity  of  the  second 
order  in  comparison  with  the  length  of  the  chord. 

The  same  results  might  have  been  established  from  the 
expansions  for  sin  a  and  cos  a,  when  a  is  considered  as  infi- 
nitely small. 

If  in  the  general  case  of  any  continuous  curve  we  take 
two  points  A,  B,  on  the  curve,  join  them,  and  draw  BE 
perpendicular  to  AB,  meeting  in  E  the  normal  drawn  to 
the  curve  at  the  point  A  ;  then  all  the  results  established 
above  for  the  circle  still  hold.  When  the  point  B  is  taken 
infinitely  near  to  A,  the  line  AE  becomes  the  diameter  of 
the  circle  of  curvature  belonging  to  the  point  A ;  for,  it  is 
evident  that  the  circle  which  passes  through  A  and  B,  and 
has  the  same  tangent  at  A  as  the  given  curve,  has  a  contact 
of  the  second  order  with  it.  See  "Salmon's  Conic  Sections," 
Art.  239. 


Examples. 

1.  In  a  triangle,  if  the  vertical  angle  be  very  small  in  comparison  with  either 
of  the  base  angles,  prove  that  the  difference  between  the  sides  is  very  small  in 
comparison  with  either  of  them ;  and  hence,  that  these  sides  may  be  regarded  as 
ultimately  equal. 

2.  In  a  triangle,  if  the  external  angle  at  the  vertex  be  very  small,  show  that 
the  difference  between  the  sum  of  the  sides  and  the  base  is  a  very  small  quantity 
of  the  second  order. 

3.  If  the  base  of  a  triangle  be  an  infinitesimal  of  the  first  order,  as  also  its 
base  angles,  show  that  the  difference  between  the  sum  of  its  sides  and  its  base 
is  an  infinitesimal  of  the  third  order. 

This  furnishes  an  additional  proof  that  the  difference  between  the  length  of 
an  arc  of  a  continuous  curve  and  that  of  its  chord  is  ultimately  an  infinitely 
small  quantity  of  the  third  order. 

4.  If  a  right  line  be  displaced,  through  an  infinitely  small  angle,  prove  that 
the  projections  on  it  of  the  displacements  of  its  extremities  are  equal. 

5.  If  the  side  of  a  regular  polygon  inscribed  in  a  circle  be  a  very  small 
magnitude  of  the  first  order  in  comparison  with  the  radius  of  the  circle,  show 
that  the  difference  between  the  circumference  of  the  circle  and  the  perimeter  of 
the  polygon  is  a  very  small  magnitude  of  the  second  order. 


4-0  Successive  Differentiation. 

38.  Fundamental  Principle  of  the  Infinitesimal 
Calculus. — We  shall  now  proceed  to  enunciate  the  funda- 
mental principle  of  the  Infinitesimal  Calculus  as  conceived  by 
Leibnitz  :*  it  may  be  stated  as  follows : — 

If  the  difference  between  two  quantities  be  infinitely 
small  in  comparison  with  either  of  them,  then  the  ratio  of 
the  quantities  becomes  unity  in  the  limit,  and  either  of  them 
can  be  in  general  replaced  by  the  other  in  any  expression. 
For  let  a,  j3,  represent  the  quantities,  and  suppose 

a  =  /3  +  *,  or  ,3=i+j3. 

Now  the  ratio  ^  becomes  evanescent  whenever  t  is  infinitely 

small  in  comparison  with  j3.  This  may  take  place  in  three 
different  ways  :  (1)  when  j3  is  finite,  and  i  infinitely  small : 
(2)  when  i  is  finite,  and  [5  infinitely  great ;  (3)  when  j3  is 
infinitely  small,  and  i  also  infinitely  small  of  a  higher  order  : 

with  /3. 


*  This  principle  is  stated  for  finite  magnitudes  by  Leibnitz,  as  follows : — 
1  i  Cseterum  aBqualia  esse  puto,  non  tantnm  quorum  differentia  est  omnino  nulla, 
sed  et  quorum  differentia  est  incomparabiliter  parva."  .  .  .  "  Scilicet  eas 
tantum  homogeneas  quantitates  comparabiles  esse,  cum  Euc.  Lib.  5,  defin.  5, 
censeo,  quarum  una  numero  sed  finito  multiplicata,  alteram  superare  potest ;  et 
qua3  tali  quantitate  non  differunt,  sequalia  esse  statuo,  quod  etiam  Arcbimedes 
sumsit,  aliique  post  ipsum  omnes."     Leibnitii  Opera,  Tom.  3,  p.  328. 

The  foregoing  can  be  identified  "with  tbe  fundamental  principle  of  Newton, 
as  laid  down  in  his  Prime  and  Ultimate  Eatios,  Lemma  I. :  "  Quantitates,  ut 
et  quantitatum  rationes,  quse  ad  sequalitatem  tempore  quovis  finito  constanter 
tendunt,  et  ante  finem  temporis  illius  proprius  ad  invicem  accedunt  quam  pro 
data"  quavis  differentia,  fiunt  ultimo  sequales." 

All  applications  of  the  infinitesimal  method  depend  ultimately  either  on  the 
limiting  ratios  of  infinitely  small  quantities,  or  on  the  limiting  value  of  the 
sum  of  an  infinitely  great  number  of  infinitely  small  quantities  ;  and  it  may 
be  observed  that  the  difference  between  the  method  of  infinitesimals  and  that  of 
limits  (when  exclusively  adopted)  is,  that  in  the  latter  method  it  is  usual  to 
retain  evanescent  quantities  of  higher  orders  until  the  em?  of  the  calculation, 
and  then  to  neglect  them,  on  proceeding  to  the  limit ;  while  in  the  infinitesimal 
method  such  quantities  are  neglected  from  the  commencement,  from  the  know- 
ledge that  they  cannot  affect  the  final  result,  as  they  necessarily  disappear  in  the 
limit. 


Principles  of  the  Infinitesimal  Calculus.  41 

Accordingly,  in  any  of  the  preceding  cases,  the  fraction 

^  becomes  unity  in  the  limit,  and  we  can,  in  general,  substi- 

tute  a  instead  of  j3  in  any  function  containing  them.  Thus, 
an  infinitely  small  quantity  is  neglected  in  comparison  with 
a  finite  one,  as  their  ratio  is  evanescent ;  and  similarly  an 
infinitesimal  of  any  order  may  be  neglected  in  comparison 
with  one  of  a  lower  order. 

Again,  two  infinitesimals  a,  j3,  are  said  to  be  of  the  same 

order  if  the  fraction  —  tends  to  a  finite  limit.     If  ¥.   tends 

a  a 

to  a  finite  limit,  j3  is  called  an  infinitesimal  of  the  nth  order 

in  comparison  with  a. 

As  an  example  of  this  method,  let  it  be  proposed  to 
determine  the  direction  of  the  tangent  at  a  point  (x,  y)  on  a 
curve  whose  equation  is  given  in  rectangular  co-ordinates. 

Let  x  +  a,  y  +  j3,  be  the  co-ordinates  of  a  near  point  on 

the  curve,  and,  by  Art.   10,  the  direction   of  the  tangent 

)3 
depends  on  the  limiting  value  of  — .     To  find  this,  we  substi- 

a 

tute  x  +  a  for  x,  and  y  +  /3  for  y  in  the  equation,  and  neglect- 
ing all  powers  of  a  and  ]3  beyond  the  first,  we  solve  for  — , 

a 

and  thus  obtain  the  required  solution. 

For  example,  let  the  equation  of  the  curve  be  x3  +  y%  =  $axy : 
then,  substituting  as  above,  we  get 

xz  +  sx2a  +  yz  +  3£/2/3  =  $axy  +  $ax$  +  $aya  : 

hence,  on  subtracting  the  given  equation,  we  get  the 

Umit  of  &  =  *—% 
a      ax  -  yA 

39.  Subsidiary  Principle. — If  ax  +  a%  +  a3  +  .  .  .  +  an 

represent  the  sum  of  a  number  of  infinitely  small  quantities, 
which  approaches  to  a  finite  limit  when  n  is  increased  indefi- 
nitely, and  if  )3i,  j32,  •  .  .  /3»  be  another  system  of  infinitely 
small  quantities,  such  that 

/3x  j32  (5n. 

—    =     I     +    £l,       —    =     I     +    fo,      ...     =     I     +    tnt 

ai  a2  an 


42  Successive  Differentiation. 

where  el9  e2,  .  .  .  £n,  are  infinitely  small  quantities,  then  the 
limit  of  the  sum  of  |3i,  j32,  .  .  .  j3w  is  ultimately  the  same  as 
that  of  ai,  a2,  .  .  .  a^. 

For,  from  the  preceding  equations  we  have 

j3i  +  j3a  +  .  .  •  + 13»  =  ai  +  a2  +  .  .  .  +  an  +  ax£i  +  a2£2  +  .  .  .  +  an£n- 

Now,  if  t}  he  the  greatest  of  the  infinitely  small  quan- 
tities, ei,  £2,  .  .  .  sn,  we  have 

j3i  +  |32  +  .  .  .  +  fin  -  (ai  +  a2  +  .  .  .  .  +  an)  <  rj  (ai  +  a2  .  .  .  +  a») ; 

hut  the  factor  ai  +  a2  +  .  .  .  +  an  has  a  finite  limit,  hy  hypo- 
thesis, and  as  rj  is  infinitely  small,  it  follows  that  the  limit  of 
fii  +  j32  +  .  •  .  +  fin  is  the  same  as  that  of  ax  +  a2  +  .  .  .  +  an. 
This  result  can  also  he  estahlished  otherwise  as  follows : — 


n 


The  ratio  P.  +  ^,+  ...  +  P> 

ai  +  a2  +  .  .  .  +  an 


hy  an  elementary  algehraic  principle,  lies  hetween  the  greatest 
and  the  least  values  of  the  fractions 

fil    p2  fin  m 

>         >  •   •   •  J 

a\    a%  an 

it  accordingly  has  unity  for  its  limit  under  the  supposed  con- 
ditions :  and  hence  the  limiting  value  of  /3i  +  fit  +  .  .  .  +  fin  is 
the  same  as  that  of  at  +  a2  +  .  .  .  +  an. 

40.  Approximations. — The  principles  of  the  Infini- 
tesimal Calculus  above  estahlished  lead  to  rigid  and  accurate 
results  in  the  limit,  and  may  he  regarded  as  the  fundamental 
principles  of  the  Calculus,  the  former  of  the  Differential,  and 
the  latter  of  the  Integral.  These  principles  are  also  of  great 
importance  in  practical  calculations,  in  which  approximate 
results  only  are  required.      For  instance,  in  calculating  a 

result  to  seven  decimal  places,  if  — j  he  regarded  as  a  small 

quantity  a,  then  a2,  a3,  &c,  may  in  general  he  neglected. 
Thus,  for  example,  to  find  sin  30' and  cos  30' to  seven  de- 

cimal  places.    The  circular  measure  of  3 o'  is  -7-,  or  .008 7  266; 

360 


Approximations.  43 

denoting  this  by  a,  and  employing  the  formulae, 


ar  er 

sin  a  =  a  -  — ,  cos  a  =  1 , 

6  2 

it  is  easily  seen  that  to  seven  decimal  places  we  have 

a2  a3 

—  =  .OOOO381,        —  =  .OOOOOOI. 
2  6 

Hence        sin  30'  =  .0087265  ;  cos  30'  =  9999619. 

In  this  manner  the  sine  and  the  cosine  of  any  small  angle 
can  be  readily  calculated. 

Again,  to  find  the  error  in  the  calculated  value  of  the 
sine  of  an  angle  arising  from  a  small  error  in  the  observed 
value  of  the  angle.  Denoting  the  angle  by  a,  and  the  small 
error  by  a,  we  have 

sin  {a  +  a)  =  sin  a  cos  a  +  cos  a  sin  a  =  sin  a  +  a  cos  a, 

neglecting  higher  powers  of  a.     Hence  the  error  is  repre- 
sented by  a  cos  a,  approximately. 

In  like  manner  we  get  to  the  same  degree  of  approxima- 
tion 

tan  (a  +  a)  -  tan  a  = 


eos2a 


Again,  to  the  same  degree  of  approximation  we  have 
a  +  a      a      ha  -  afi 

where  a,  j3  are  supposed  very  small  in  comparison  with  a  and  b. 
As  another  example,  the  method  leads  to  an  easy  mode  of 
approximating  to  the  roots  of  nearly  square  numbers  ;  thus 

\/a2  +  a  =  a  +  —  ;    ^/a2  +  a2  =  a  +  —  =  a,  whenever  a2  may 

2  Co  2,Ct 

be  neglected. 

Likewise,  l/az  +  a  =  a  +  — =,  &o. 

If  b  =  a  +  a,  where  a  is  very  small  in  comparison  with  a, 

i  /— 7       /-= a      a  +  b 

we  nave         */ab  =  yar  +  aa  =  a  +  -  = . 


44  Successive  Differentiation, 

Again,  in  a  plane  triangle,  we  have  the  formula 

C  C 

c2  =  a2  +  b2  -  2ab  cos  C  =  (a  +  b)2  sin2  —  +  (a  -  b)2  cos2  — . 

Now  if  we  suppose  a  and  b  nearly  equal,  and  neglect  (a-b)2 
in  comparison  with  {a  +  b)z,  we  have 

/  O  C  C 

c=    Ua  +  b)2  sin2  —  +  (a  -  b)2  cos2  —  =  {a  +  b)  sin  — . 

This  furnishes  a  simple  approximation  for  the  length  of 
the  base  of  a  triangle  when  its  sides  are  very  nearly  of  equal 
length. 

Exaiuples. 

i.  Find  the  value  of  (r  +  a)  (i  -  2a2)  (r  +  3a3),  neglecting  a*  and  higher 
powers  of  a.  Ans.  I  +  a  —  2o2  +  a3. 

2.  Find  the  value  of  sin  {a  -t-  o)  sin  (b  +  j6),  neglecting  terms  of  2nd  order 
in  a  and  j8.  Am.  sin  a  sin  6  +  a  cos  a  sin  5  +  fi  sin  a  cos  b. 

3.  If  m  =  u  —  e  sin  w,  e  heing  very  small,  find  the  value  of  tan  § u. 

AVI 

Ans.  (1  +  e)  tan  — . 


tan  -  =  tan  I ho),  where  a  =  -smw;  .\  &c. 

2  \2  /  2 


u      m      e    . 
Here  -  =  —  4-  -  sin  u : 
222 

4.  In  a  right-angled  spherical  triangle  we  have  the  relation  cos  c  —  cos  a  cos  b; 
determine  the  corresponding  formula  in  plane  trigonometry. 

The  circular  measure  of  a  is  — ,  R  being  the  radius  of  the  sphere ;  hence, 

jS 

a2 

substituting  1 for  cos  a,  &c,  and  afterwards  making  R  =  00,  we  get 

R2 

c2  =  a2  +  b2. 

5.  If  a  parallelogram  be  slightly  distorted,  find  the  relation  connecting  the 
changes  of  its  diagonals. 

Ans.  dAd  +  d'Ad*  =  o,  where  d,  d'  denote  the  diagonals,  and  Ad,  Ad'  the 
changes  in  their  lengths.  In  the  case  of  a  rectangle  the  increments  are  equal, 
and  of  opposite  signs. 

6.  Find  the  limiting  value  of 

j_am  +  £am+1  +  Cam+2  +  &c. 

aan  +  ban+1  +  ca™2  +  &c. 

when  a  becomes  evanescent. 

.  Aam       A 
In  this  case  the  true  value  is  that  of  — —  =  —  aw_n. 

aan         a 

A 
Hence  the  required  value  is  zero,  — ,  or  infinity,  according  as  m>,  =,  or  <  n, 

a 


Examples.  45 


7.  Find  the  value  of 


xz       x* 

1--  +  — 

6       120 


x2      x^ 
1 +  — 

2       24 

neglecting  powers  of  #  beyond  the  4th.  -4ws.  1  H h  - — . 

8.  Find  the  limiting  values  of  -  when  y  =  o,  x  and  y  being  connected  by 

y 

the  equation  «/3  =  2xy  —  a;2. 
Here,  dividing  by  y%  we  get 

#2         # 

—  -  2  -  =  -  y. 

y*      y 


If  we  solve  for  -  we  have 


5-.±d-rtt 

Hence,  in  the  limit,  when  «/  =  o,  we  have  -  =  2,  or  -  =  o. 

y  y 

9.  In  fig.  3,  Art.  37,  HAB  be  regarded  as  a  side  of  a  regular  inscribed  polygon 
of  a  very  great  number  of  sides,  show  that,  neglecting  small  quantities  of  the 
4th  order,  the  difference  between  the  perimeter  of  the  inscribed  polygon  and 
that  of  the  circumscribed  polygon  of  the  same  number  of  sides  is  represented 

by  -  BD. 

Let  n  be  the  number  of  sides,  then  the  difference  in  question  is  n  (AD  —  AB) ; 

,    A  ttAE  ,^       ,„,     v  AE (AD  -  AB) 

but  n= — ;     .-.  n(AD-AB)=  1— * 

arc  AB  AB 

T)J? A  J?  v 

=  *AE         ."T     =  w(DE-AE)  =  -  BD,  q.  p. 
AE  2 

This  result  shows  how  rapidly  the  perimeters  of  the  circumscribed  and  in- 
scribed polygons  approximate  to  equality,  as  the  number  of  sides  becomes  very 
great. 

10.  Assuming  the  earth  to  be  a  sphere  of  40,000,000  metres  circumference, 
show  that  the  difference  between  its  circumference  and  the  perimeter  of  a  regular 
inscribed  polygon  of  1,000,000  sides  is  less  than  r§-th  of  a  millimetre. 

11.  If  one  side  b  of  a  spherical  triangle  be  small,  find  an  expression  for  the 
difference  between  the  other  sides,  as  far  as  terms  of  the  second  order  in  b. 

Here  cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  G. 

Let  a  denote  the  difference  in  question ;  i.  e.  e  =  a  —  z ; 

then  cos  a  cos  z  +  sin  a  sin  z  =  cos  a  cos  b  +  sin  a  sin  b  cos  G; 

,\  sin  z  -  sin  b  cos  G  =  cot  a  (cos  b  —  cos  z). 


4 6  Successive  Differentiation. 

Since  z  and  b  are  both  small,  we  get,  to  terms  of  the  second  order, 
z  -  b  cos  G  = (z2  -  b2). 

2 

The  first  approximation  gives  z  =  b  cos  C.     If  this  be  substituted  for  z  in  the 
right-hand  side,  we  get,  for  the  second  approximation, 

_     £2  sin2  C  cot  a 
z  =  b  cos  C . 

2 

We  now  proceed  to  find  the  successive  derived  functions 
in  some  elementary  examples. 

41.  Derived  Functions  of  xm. 

Let  y  =  xm, 

.,  dy  <py         . 

then  —  =  mx™"1,  ■—  =  m[m  -  1)  xm~2, 

UX  UX 

and  in  general,  j^n  =  m(m-  i)  (m  -  2)  .  .  .  (m  -  n  +  1)  x™-™. 

If  m  be  a  positive  integer,  we  have 

dm  (xm) 


dxr 


-  1 • 2  ...  m. 


and  all  the  higher  derived  functions  vanish. 

If  m  be  a  fractional,  or  a  negative  index,  then  none  of  the 
successive  derived  functions  can  vanish. 


Examples. 
1.  If  u  =  ax™  +  bxn~l  +  ex™"2  +  &c,  prove  that 

—  =  n  (n  —  1)  ax71'2  +  (n  -  i)  (n  -  2)  bxn-2  +  &c, 

dnu  ,  dn+1u 

also  ——  =  1  .  2  .  .  .  .  n  .  a,  and  - — ,  =  o. 

dx*  dxn+1 


2. 


prove  that 


dy         na,       d2y     n(n  4-  i)a 
dx~~~  %n+V     (fat  ~       ^£2      » 


,                          <Zmv      ,      ,     n(n+i) ...  (n+m-i)a 
and  -r-^  =  (-  iV»  — - i - — . 

dxm        s        '  %n+m 


Examples,  47 

dy         a        dry  _        a       dzy  _  3  a 
dx  ~  */#'     dx%  2#§'     dxd      4  #|' 

«^  =  /_  j\„  3-5-7  ••  •  (aw~i)a  _ 

d^y 
42.  If 2/  =  xz  log  a?,  to  find  -— f . 

Here  —  =  3a?2  log  #  +  a?2 ; 

also  —  =  6,2?  log  x  +  $x  +  20  =  6,1?  log  a;  +  $x, 

CIX 

d3y         .  <tf4v      6 

^=61og*+6  +  5,      ^=-. 

It  might  have  been  observed  that  in  this  case  all  the 

terms  in  the  successive  differentials  which  do  not  contain 

log  x  will  disappear  from  the  final  result — thus,  by  the  last 

ds  (x2) 
Article,  '  =  o,  accordingly,  that  term  may  be  neglected  ; 

ux> 

and  similar  reasoning  applies  to  the  other  terms.  The  work 
can  therefore  be  simplified  by  neglecting  such  terms  as  we 
proceed. 

The  student  will  find  no  difficulty  in  applying  the  same 
mode  of  reasoning  to  the  determination  of  the  value  of 

-=---,  where  y  =  xn~x  log  x. 

For,  as  in  the  last,  we  may  neglect  as  we  proceed  all  terms 
which  do  not  contain  log  a?  as  a  factor,  and  thus  we  get  in 
this  case, 

dny  _  (n  -  1)  .  .  .  2  .  1  _  \n~  I 
dxn  x  x 


48  Successive  Differentiation. 

43.  Derived  Functions  of  sin  mx. 

Let  y  =  sin  mx, 

then 


dy 
dx 

d2y 


—  =  m  cos  mx, 
dx 


dx 


=  -  m*  sm  mx, 

A  * 


d2ny 
and,  in  general,  -j~  =  (-  i)nm2n  sin  mx, 
dx 


~\ 


d2n+1y 


dx2n+ 


7  =  (-  i)nm2n+1  cos  mx. 


<o 


It  is  easily  seen  that  these  may  be  combined  in  the  single 
equation  (Art.  22), 


dr  (sm  mx)        m   . 

— ^-z =  mr  sm  (  mx  +  r  - 

dxr 


(*) 


7T 

m'  cos  I  mx  +  r  -  1  • 


In  like  manner  we  have 
dr  cos  m# 

44.  Derived  Functions  of  eax. 
Let  y  =  ea% 
dy 


then 


=  «e° 


d2?/       ,  dny 

^  =  a2eaz,  .  .  .  -r~  =  «"* 


(3) 


(4) 


where  the  symbol  ( ~ )  denotes  that  the  process  of  differentia- 
tion is  applied  n  times  in  succession  to  the  function  eax. 


dx  dx2  '  dxn 

This  result  may  be  written  in  the  form 

d\n 

pUX    _    rMpCLX 


dx 
d\n 


Derived  Functions  of  eax  cos  bx.  49 

In  general,  adopting  the  same  notation,  we  have 

=A0aneax  +  AxdP"1*?*  +  A%an~%eax  +  &c. 
=  [^o«w  +  ^i^'1  +  -^s^"2  +  &o.  -4  J  eaa;. 
This  result,  if  #  (x)  denote  the  expression 
Adcn  +  Axxn~l  +  ...  An, 
may  be  written  in  the  form 

*  (J;)  «"•  =  *(*)*"*;  (5) 

in  which  $  (a)  is  supposed  to  contain  only  positive  integral 
powers  of  a. 

45.  To  find  the  nth  Derived  Function  of  eax  cos  bx. — 

Let  y  represent  the  proposed  expression, 

d'fj 
then  -~  =  aeax  cos  bx  -  beax  sin  bx 

ax 

-  eax  (a  cos  bx  -  b  sin  bx) ; 

if  tan  <p  =  -,  we  have  b  =  */a%  +  b2  sin  $,  and  a  =  v^3  +  bz  cos  0, 
Hence  we  get 

^L  =  (a»  +  §2)|  ^w  cos  (^  +  ^% 


50  Successive  Differentiation. 

Again, 

_|  =  (a?  +  b2)i  eax  \_a  cos  (bx  +  0)  -  b  sin  (£#  +  0)] 

=  (a2  +  b2)  eax  cos  (5a?  +  20). 

By  repeating  this  process  it  is  easily  seen  that  we  have  in 
general,  when  n  is  any  positive  integer, 

yt  =  (a2  +  b2)2  eax  cos  {bx  +  n<£).  (6) 

(JjX 

46.  To   find  the  Derived  Functions   of  tan  -1  ( -  j, 

and  tan-1  x. 

Let  y  =  tan"1  ( -  ],  or  x  =  cot  y : 

\Xj 

,'  dy       -  1  .  „ 

then  -7-  = 5  =  -  sin2  y. 

ax      1  +  or 

d2y       d  fdy\         d  ,  .  _    .  dy  d  ,  .  _   x 

_f  =  _  -f  =  -  _  (sm2  y)  =  -  -f  —  (sm2*/) 
c&r      a#  \a^/        «■£  ax  ay 

=  sin2  y  -7-  (sin2  ?/)  =  sin2  y  sin  2^. 

t/ 

.      .  dzy      d   .  .  .  dy  d  . 

A^ain'    «; =  & (sin  y  sm  2y)  =  &  * (sm  y  sm  2y) 

=  -  sin2  y  —  (sin2  y  sin  2«/) 
=  -1.2.  sin3  y  sin  3^.  (.Ek.  5,  Art.  28.) 

Hence,  also  —  =  1.2.3.  ™  y  sm  4^ ; 


^4 
dx 


dnij 
and  in  general,  —  .=  (-  i)w  |»  -  1  $mny  sin  ny. 


Theorem  of  Leibnitz.  51 

IT  I 

Again,  since  tan-1  x  =  —  tan-1  -, 

2  (v 

we  nave      — ^ — -  =  (-  i)%  x  ^  -  1  sinK  y  sm  ny,  (7) 

where  2/  =  cot-1#,  as  before. 

This  result  can  also  be  written  in  the  form 

d '(tan'o)  _  ,_    .M  V  "/ .  (8) 

47.  If  y  =  sin  (m  sin"1*),  to  prove  tbat 

d~ii         du 

Here 

dy     m  cos  (m  sin""1  x) 

dx  y/I  _  ^2 

.•.  (1  -  a?2)  (  —  J  =  m?  cos2  (m  sin-1#)  =  m2  (1  -  ?/2). 

/My 

Hence,  differentiating  a  second  time,  and  dividing  by  2  — , 
we  get  the  required  result. 

48.  Theorem  of  IJeibnitz. — To  find  the  nth  differen- 
tial coefficient  of  the  product  of  two  functions  of  x.  Let 
y  =  uv;  then,  adopting  the  notation  of  Art.  34,  we  write 

,     ,     ,   p     dy  du        _  dv 

and  similarly,  y'\  u",  v",  &c,  for  the  second  and  higher 
derived  functions — thus, 

E  2 


52  Successive  Differentiation. 

Now,  if  we  differentiate  the  equation  y  =  uv,  we  have 

y'  =  uv'  +  vu',  by  Art.  13. 
The  next  differentiation  gives 

y"  =  uv"  +  u'v'  +  v'u'  +  Vu"  =  UV"  +  2u'v'  +  Vu". 

The  third  differentiation  gives 

if"  :=  uv"'  +  U'v"  +  2lt'v"  +  2U"V'  +  V'U"  +  Vu'" 

=  uv"'  +  su'v"  +  $u"v'  +  vu"\ 

in  which  the  coefficients  are  the  same  as  those  in  the  expan- 
sion of  (a  +  by. 

Suppose  that  the  same  law  holds  for  the  nth  differential 
coefficient,  and  that 

yW  =  uvW  +  nu'v^1-1)  +  -± '-  u"v  M  +  &c, 

+  nu^V  v'  +  w(")  v  ; 
then,  differentiating  again,  we  get 

y&+l)  =  uv^1)  +  u'vW  +  n  (u'vW  +  u"v(n-V) 
+  n(n~  ')  {u"v(n~l)  +  u"'v^-*))  +  &c.  .  .  .  +  u^v 

=  uv^1)  +  (n  +  1)  u'vW  +  lw+  ^VcW  +  &c.  .  .  . , 

1.2 

in  which  it  can  be  easily  seen  that  the  coefficients  follow  the 
law  of  the  Binomial  Expansion. 

Accordingly,  if  this  law  hold  for  any  integer  value  of  n, 
it  holds  for  the  next  higher  integer  ;  but  we  have  shown  that 
it  holds  when  n  =  3  ;  therefore  it  holds  for  n  =  4,  &c. 

Hence  it  holds  for  all  positive  integer  values  of  n. 

In  the  ordinary  notation  the  preceding  result  becomes 

dn(uv)        dnv        dudn~lv     n(n  -  1)  d2udtl~2v      0 
dxn  dxn        dx  dxn~l  1.2     d&  dxn~2 

it  u  /      \ 


Applications  of  Leibnitz's  Theorem.  53 

49.  To  prove  that 

d\n  f         d\n 

—     (eax u)  =  eax [a  +  —  )  u,  (11) 

dx)   v  \        dx) 

where  n  is  a  positive  integer. 

Let  v  =  eax  in  the  preceding  theorem  ;  then,  since 


dv  d?v       „  dnv       m  „„ 

4-  =  aeax     —  =  aV*,  .  .  .  —  =  aneax, 


we  have 


\dx)  x  \  <fe        1  .  2         •  dx2  dxnJ 

which  may  be  written  in  the  form 

d\n  {  d    n(n-i)      J  d\    0       ( d\n) 

-    (eaxu)  =  eax \an  +nan~1—+^ V"2  — )  +  &c.+  —    \u, 

dx j  (  ax      1.2  \ax j  \axj  j 


( d\n  ,       x  /         d\n 

(s)(^)-<-(«.+  5)«; 

/        d\n  . 
where  the  symbolic  expression  f #  +  —  \  is  supposed  to  be 

,  1  1     1         j  i        t>  •  •    i     mi  n    ^^      $  W  tt  W 

developed  by  the  -binomial  Theorem,  and  — ,  — ,  .  .  .  — 

ax   ax  ax 

substituted  for  ( —  j  u,  ( —  J  u,  [-f\  u,  in  the  resulting  ex- 
pansion. 

50.  In  general,  if  <j>(a)  represent  any  expression  in- 
volving only  positive  integral  powers  of  a,  we  shall  have 

For,  let  (j)  ( — ),  when  expanded,  be  of  the  form 


d_\n        /dy-1 

dx)  l  \dx^ 


A0[—)+A1[  —  )     +  .  .  .  +  An, 


54  Successive  Differentiation. 

then  the  preceding  formula  holds  for  each  of  the  component 
terms,  and  accordingly  it  holds  for  the  sum  of  all  the  terms  ; 
.'.  &c. 

The  result  admits  also  of  being  written  in  the  form 

This  symbolic  equation  is  of  importance  in  the  solution 
of  differential  equations  with  constant  coefficients.  See 
"  Boole's  Differential  Equations,"  chap.  xvi. 

51.  Iff  y  =  sin-1  x,  to  prove  that 

/       2^  dn+2y    f        x    dMly      .dny 

Here  4~  = — ?         or  (1  -  #2)i-^  =  1 ; 

dx      </i  -  x2  dx 

hence,  by  differentiation, 

x^ 

Kl     X)°dx*      (1  -  x>)l  -  °' 

Again,  by  Leibnitz's  Theorem,  we  have 

d\n{,         ^dry)      r         ^dn+2y  dn+1y         .         ,dny 

—      (1  -x2)  -rr-Jt  =  (1  -x2)-—£--  2nx-~  ~n(n-  i)~-. 
dx)  (v  }  dx2)      K  Jdxn+2  dxM1         v  J  dxn 

( d\n  (   dy)        dn*ly        dny 
Also  —     \as-f  \=x-—^-^n-\. 

\dx)    (   dx)         dxn+1         dxn 

On  subtracting  the  latter  expression  from  the  former,  we 
obtain  the  required  result  by  (14). 
If  x  =  o  in  formula  (13),  it  becomes 


\dxna)0         \dxnJ«       ' 


Applications  of  Leibnitz's  Theorem.  55 

'*y\  *-  iL  ~iJ:  ^  fry 


where  f  —  j   represents  the  value  of  —^  when  x  becomes 

cypher. 

Also,  since  (—  j   =  1,  we  get,  when  n  is  an  odd  integer, 

\daf»%Jo  d      5       '  '  *      " 

Again  we  have  ( —  J  =  o ;  consequently,  when  n  is  an  even 

integer,  we  have  (^  =  a 

Ml 

52.  If  ^  =  (1  +  #2)2sin  (wtan"1^),  to  prove  that 

^  +    >dx~2~  2(m  ~  0*;^ +  "»(*»-  0y  =  o.    (15) 

Here 

d?y  --1  .  --1 

—  =  m#(i  +  a;2)2     sin(m  tan*"1  a?)  +  m(i  +  x2)2   cos  (m tan"1  a?), 

or 

(1  +  a;2)  —  -vnx(\  +  a;2)2sin(mtan~1^)  +  w(i+ii?2)2cosm(tan~1^) 

m 

=  mxy  +  m(i  +  x2)2  cos  (m  tan"1  x) ; 

.'.  (1  +  x2) 2  cos  (m  tan"1  a?)  = f  -  xy. 

x  y         v  '        m     dx       * 

The  required  result  is  obtained  by  differentiating  the  last 
equation,  and  eliminating  cos  (m  tan-1  x)  and  sin  [m  tan"1^)  by 
aid  of  the  two  former. 

Again,  applying  Leibnitz's  Theorem  as  in  the  last  Article, 
we  get,  in  general — 

/      2N  dn+zy    r  n  dn+1y  t      x  /  \&y 


56  Successive  Differentiation, 

Hence,  when  x  =  o,  we  have 

Moreover,  as  when  x  =  o,  we  have  y  =  o,  and  ~  =  m  ;  it 
follows  from  the  preceding  that 

M)0=O;  MJo=  ("  0"*(«-i)  .  .  .  (—**    (16) 

For  a  complete  discussion  of  this,  and  other  analogous 
expressions,  the  student  is  referred  to  Bertrand,  "  Traite  de 
Calcul  Differ entiel,''  p.  144,  &c. 


Examples,  57 


Examples. 
1.  y  =  «*  log  a?,                                    prove  that  ^  =- *=;• 
2.y  =  x\ogx.  „        ■tZ=(-i)w r • 

3.  y  =  x*,  „        —  =  ^(i  +  log»)2  +  ^1. 

<?3y      2  cos  X 

4.  y- log  (am*),  „        ;p=-^- 

>V^  1  +  jb2  - 1  2*  rf  V  5# 

5.y  =  tan-i +  tani— ,     „        _  =  _____ 

6.  y  =  s«log(s§),  „        ^5-=-- 


'• ' = log V ,-,^+^       ^ "     ^  =  " F^¥" 

where  tan  <p  =  -. 
r 

9.  If  y  =  tf0**'",  prove  that 

*?M«/  ,       ,      w(w  -  i)r  (r  -  1)       .      „  "I 

— -  =  eax  \  anxr  +  nran"1  x*~i  +  — — ± an~z  xr~2  +  .  .  .    , 

dxn  1.2  J 

„d  (s)V^-g)*,(5)"(-*0. 

10.  If «/  =  a  cos  (log  #)  ■+-  b  sin  (log  a:), 

prove  that  x2  —■  +  x  ,   +  y  =  o. 

tfos         dx 

11.  If  2/  =  ef* sin_la;, 

,-.    ,  ,         .,  d2y        dy 

prove  that  (1  -  a2)  —  -  a;  -  =  a2y. 


58  Examples. 

12    Prove  that  the  equation 

_.  d2y        dy       „ 

is  satisfied  by  either  of  the  following  values  of  y  : 

y  —  cos  (0  sin-1  #),  or  y  =  ea  _1  an-i*. 

13.  Being  given  that        y  =  (#  +  v  a;2  —  i)OT, 

,    .  ,  „       .  d2y         dy        _ 

prove  that  (a;2  —  1)  — —  +  a;  - —  m2y  =  o. 

dx2         dx 

14.  If  y  =  sin  (sin  #), 

prove  that  — -  +  —  tan  a;  +  y  cos2#  =  o. 

(ajOO       doc 

15.  In  Fig.  3,  Art.  37,  if  AB  be  regarded  as  a  side  of  a  regular  polygon  of  an 
indefinitely  great  number  of  sides,  show  that  the  difference  between  the  circum- 

ference  of  the  circle  and  the  perimeter  of  the  polygon  is  represented  by  ^  BD, 

to  the  second  order  of  infinitesimals. 

/  d<l         \ 

16.  If  y  =  A  cos  nx  +  B  sin  nx,  prove  that  I  —  +  n2\  y=?o. 

1  ,    ,  dny      ,      .    \n.sm.n+1<b  sm(n  +  i)d> 

17.  If  y  = ,  prove  that  ■—  =  (1  -)»  '-  y       v         /y 

where  d>  =  tan*1  -. 
x 

This  follows  at  once  from  Art.  46,  since  —  f  tan-1-  J  = ...    It  can  also  be 

dx  \         xj       a2  +  xz 

proved  otherwise,  as  follows  : 

_i_=_i_r_j l_1- 

a2  +  x2      20  (-  if  \_x  -  0  (-  i)£     a;  +  0  (-  1)* J  ' 

^»«/  _         1         /  d  \ »  1  1  /  0*  \ »  1 

dxn  ~  2a  (-  i)i  \dx)    '  x  —  0  (—  1)*      2a  (-  1)*  V027    "  a;  +  0  (-  1)* 

_  (-  i)«  I  .  2  ...»  [~  1  1  "1 

=         20  (-  ijl  |_(«  -0(-i)^+i  "  (a?  +  0  (-  i)*)M+1  J 

_(-i)»|»  |"(a;  +  0  (-  i)*)"*1)  -  (a;  -  0  (-  i)*)™*"] 
"  27(=l)i  L~  WTa^  J ' 


Examples.  59 

Again,  since  -  =  tan  <b,  we  havo  a  =  \Za2  -t-  x2  sin  <p,  and  x  =  ya2  +  x2  cos  ty  ; 
x 

n+  1 

hence  (x  +  a(-  i)iy»l  -  {a2  +  x2)  2  (cos<£  +  (-  i)*  sin  $)n+x 

n  +  \ 

=  {a2  +  x2)  2  {cos(ra  +  1)  $  +  (-  i)*  sin  (»  +  1)  ^}, 

and  we  get,  finally, 

jri*.  \n .  sin  (w  +  1)  <j> .  sinw+1  $ 

— 2  -  (_  i)n  tr , 

<&;«      v      '  «w+3 

18.  In  like  manner,  if  y  =  — -, 

a1  +  x~ 

dny  \n  .  sin'l+1  <$> .  cos  (n  +  1)  <f> 

prove  that  — -  =  (-  i)»  — . 

dxn  an+1 

19.  If  u  =  xy, 

,.    ,  dnu        dny        dn~^y 

prove  that  -—  =  x  — -  +  n 


dxn        dxn        dxn~1' 
20.  If  u  —  (sin"1  x)2, 

prove  that  (1  -  x 2)  —r  -  a;  —  =  2. 

21.  Prove,  from  the  preceding,  that 

and  (*5fl   _W**\  . 

22.  If  y  =  *«*  sin  bx,  prove  that  — ^  -  zct  —  +  (a2  +  b2)  y  =  o. 

dx2  dx 


ax  +  b  dny 

-z -9,  nnd  - — . 

x*  —  c2  dxn 


23.  Given  y  =  - ,  find  — -. 


TT  ax  +  b      ac  +  b     I  etc  -  b     I 

Here = 1 . 

x2—  c2         ic     x  -  c         ic     x  +  c 

Hence  *SL  =  t_^l-  (JLtL  +  -  — *   ^ 

^W  2C  \(X  -  C)n+1        (X  +  tf)»+1/  * 


(     6o     ) 


CHAPTEE  III. 

DEVELOPMENT    OF    FUNCTIONS. 

53.  Lemma. — If  u  be  a  function  of  x  +  y  which  is  finite 
and  continuous  for  all  values  of  x  +  y,  between  the  limits 
a  and  b,  then  for  all  such  values  we  shall  have 

du     du 
dx      dy 

For,  let  u  =f(x  +  y),  then  if  x  become  x  +  h, 

—  =  limit  of^"*-^*7*)  ~f(x  +  y) 
dx  h 

when  h  is  infinitely  small. 

Similarly,  if  y  become  y  +  h,  we  have 

— = limit  of  f(x  +  y  +  h)  -f(x  +  y) 

dy  h  ' 

which  is  the  same  expression  as  before. 

-r-r  du     du 

Hence  —  =  — . 

dx      dy 

Otherwise  thus  : — Let  z  =  x  +  y,  then  u  =/(s), 

dz  dz 

dx       '  dy       ' 

du     du  dz        ,.  . 
dx     dz  dx 

du      dudz       „,.  .      du 

-7-  =  -v  ~r  =  J  M  =  ~T' 
dy      dz  dy  dx 


Taylor' }s  Expansion.  61 

54.  If  f(x  +  y)  be  a  continuous  function,  which  does  not 
become  infinite  when  y  =  o,  its  expansion  in  powers  of  y  can 
contain  no  negative  powers  ;  for,  suppose  it  contains  a  term  of 
the  form  My~m,  where  M  is  independent  of  y,  this  term  would 
become  infinite  when  y  =  o  ;  but  the  given  function  in  that 
case  reduces  to  fix)  ;  hence  we  should  have /(a?)  =  co,  which 
is  contrary  to  our  hypothesis.  Consequently  the  expansion 
of  f{x  +  y)  can  contain  only  positive  powers  of  y. 

Again,  iif(x)  and  its  successive  derived  functions  be  finite 
and  continuous,  the  expansion  of  fix  +  y)  can  contain  no 
fractional  power  of  y.    For,  if  it  contain  a  term  of  the  form 

Pyn+q,  where  -  is  a  proper  fraction,  then  its  (n  +  i)th  derived 

-* 
function  with  respect  to  y  would  contain  y  with  a  negative 

index,  and,  accordingly,  would  become  infinite  when  y  =  o  ; 
which  is  contrary  to  our  hypothesis. 

Hence,  with  the  conditions  expressed  above,  the  expan- 
sion of  f(x  +  y)  can  contain  only  positive  integral  powers  of  y. 

55.  Taylor's  Expansion  of/ (x  +  y).* — Assuming  that 
the  function /(#  +  y)  is  capable  of  being  expanded  in  powers  of 
y,  then  by  the  preceding  this  equation  must  be  of  the  form 

f(x  +  */)  =  P0  +  P<y  +  P*f  +  &g.  +  Pnif  +  &c, 

in  which  P0,  P1}  .  .  .  Pn  are  supposed  to  be  finite  and  con- 
tinuous functions  of  x. 

When  y  =  o,  this  expansion  reduces  to  fix)  =  P0. 

Again,  let  u  =f(x  +  y) ;  then  by  differentiation  we  have 

du      dP0        dPx      AP%  dPn 

-r  =  —r-  +  y—r-+y  —r~  +  •  •  •  +  V  ~r~  +  &c- ; 
dx       dx  dx  dx  dx 

—  =  Pi  +  2P2y  +  $P3yz  +  &g. 

*  The  investigation  in  this  Article  is  introduced  for  the  purpose  of  showing 
the  beginner,  in  a  simple  manner,  how  Taylor's  series  can  be  arrived  at.  It  is 
based  on  the  assumption  that  the  function  f{x  +  y)  is  capable  of  being  expanded 
in  a  series  of  powers  of  y,  and  that  it  is  also  a  continuous  function.  It  demon- 
strates that  whenever  the  function  represented  by/(#  +  y)  is  capable  of  being 
expanded  in  a  convergent  series  of  positive  ascending  powers  of  y,  the  series 
must  necessarily  coincide  with  the  form  given  in  (1).  An  investigation  of  the 
conditions  of  cOnvergency  of  the  series,  and  of  the  applicability  of  the  Theorem 
in  general,  will  be  introduced  in  a  subsequent  part  of  the  Chapter.  The  parti- 
cular case  of  this  Theorem  when  f(x)  is  a  rational  algebraic  expression  of  the  nth 
degree  in  x  is  already  familiar  to  the  student  who  has  read  the  Theory  of  Equations. 


62  Development  of  Functions. 

Now,  in  order  that  these  series  should  be  identical  for  all 
values  of  y  the  coefficients  of  like  powers  must  be  equal. 
Accordingly,  we  must  have 


_dP0_df(x) 
rx  ~  dx  -     dx     ~f  {)' 

p          i     dPx         i     d2f(x) 

I    .    2    dx          1.2      £&£3             I 

— 2/»> 

p          I  ^P2                I           d3/(tf) 

3  cfo;       i  .  2  .  3     dxz         i 

.;v"(*); 

and  in  general, 

i         ^/(a>)              i 

I  .  2  ...  91       dxn             1.2... 

Accordingly,  when  /(a?)  and  its  successive  derived  func- 
tions are  finite  and  continuous  we  have 

/(•+»)  -/(•) + f  /» + j^rw + •  •  • + £/«  w+.  •  •  (i) 

This  expansion  is  called  Taylor's  Theorem,  having  been  first 
published,  in  17 15,  by  Dr.  Brook  Taylor  in  his  Hethodus 
Incrementorum. 

It  may  also  be  written  in  the  form 

x     *t  x     yclfix)       if    d2f(x)  yn  dnf(x) 

y  v      *'    ^  w     1    da>         1 . 2     ^2  |  to      daj»         ' w 

or,  if  m  =  /(a?),  and  toi  =  f(%  +  to), 

toc?to       to2    d2u  yn  dntt      .  ,  . 

i&      1.2  da?  I  to  dkw 

To  complete  the  preceding  proof  it  will  be  necessary  to 
obtain  an  expression  for  the  limit  of  the  sum  of  the  series 
after  n  terms,  in  order  to  determine  whether  the  series  is 
convergent  or  divergent.  We  postpone  this  discussion  for 
the  present,  and  shall  proceed  to  illustrate  the  Theorem  by 


The  Logarithmic  Series.  63 

showing  that  the  expansions  usually  given  in  elementary- 
treatises  on  Algebra  and  Trigonometry  are  particular  cases 
of  it. 

56.  The  Binomial  Theorem. — Let  u  =  (x  +  y)n  ; 

here/(#)  =  xn,  therefore,  by  Art.  41, 

f{x)  =  nxn~\  .  .  .  /W  (w)  =  n(n  -  1)  .  .  .  (n  -  r  +  1)  xn~r. 
Hence  the  expansion  becomes 

(0  +  ^)»  =  af»  +  -  xn~ly  +  — xn~Y  +  •  •  • 

I  .  2  .  .  .  r  u 

If  w  be  a  positive  integer  this  consists  of  a  finite  number  of 
terms ;  we  shall  subsequently  examine  the  validity  of  the 
expansion  when  applied  to  the  case  where  n  is  negative 
or  fractional. 

57.  The  logarithmic  Series. — To  expand  log  (x  +  y). 

Here       /(*)  =  log  .(*),    /»=A    /"M  —  i 

/»  = i  •  •  •  /« w  -  (-  «r  '•'•••(*-I>. 

Accordingly 

i     /       \    i  y     I  p2     I  yz    ^  yi     o 

log  (a?  +  y)  =  log  x  +  £  -  -J-  +-V-T^  &o. 
'  «      2r      3  #3     4  #4 

If  a?  =  1  this  series  becomes 

log(I+2/)=f-f  +  ^-...(-i)'-1J..&e.  (5) 

When  taken  to  the  base  0,  we  get,  by  Art.  29, 

iog.(i+y)-.ar(|-£  +  £-£  +  &o.)-  (6) 


64  Development  of  Functions. 

58.  To  expand  sin  (x  +  y). 

Here  f{x)  =  sin#,      f{%)  =  cos  a?, 

f\x)  =  -  sin  x,     f\x)  =  -  cos  x,  &c. 
Hence 

sin  (a;  +  y)  =  sin  #   1  -  -^—  + &c.  ±  -. — 

v         '  \       1.2      1.2.3.4  \2n 

(y       yz  y5  y2n'x        \ 

+  cos  x[J- - —  + .  ..±  — .  (7) 

\I     1.2.3     1.2.3.4.5 


in 


As  the  preceding  series  is  supposed  to  hold  for  all  values, 
it  must  hold  when  x  =  o,  in  which  case  it  becomes 


#3  nfi 


sin  y  =V 1 —  + 1 &c.  (8) 

I     1.2.3     1.2.3.4.5 

7T 

Similarly,  if  x  =  — ,  we  get 

cos  y  =  1  — - —  + &c.  (9) 

9  1.2      1.2.3.4 

We  thus  arrive  at  the  well-known  expansions*  for  the  sine 
and  cosine  of  an  angle,  in  terms  of  its  circular  measure. 

59.  Maclauriu's  Theorem. — If  we  make  x  =  o,  in 
Taylor's  Expansion,  it  becomes 

/  (jf)  =/(o)  +  f/(o)  +  -^-/"(o)  +  .  .  .  £/«(<>)  +  ..'.,  (10) 

where /(o)  .  .  ,/W(o)  represent  the  values  which  f(x)  and 
its  successive  derived  functions  assume  when  x  =  o. 

Substitute  x  for  y  in  the  preceding  series  and  it  becomes 

/(*)  =/(o)  +  X-  /'(o)  +  f-  /"(o)  +  ...+£  /W  (°)  +  &c- 

1  1    .  £  \iti 


*  These  expansions  are  due  to  Newton,  and  -were  obtained  by  him  by  the 
method  of  reversion  of  series  from  the  expansion  of  the  arc  in  terms  of  its  sine. 
This  latter  series  he  deduced  from  its  derived  function  by  a  process  analogous 
to  integration  (called  by  Newton  the  method  of  quadratures).  See  Opuscula, 
torn  1.,  pp.  19,  21.  Ed.  Cast.    Compare  Art.  64,  p.  68. 


Exponential  Series.  65 

This  result  may  be  established  otherwise  thus ;  adopting 
the  same  limitation  as  in  the  case  of  Taylor's  Theorem : — 

Assume    f(x)    =  A  +  Bx  +  Ox%  +  Bxz  +  Ex"  +  &c. 
then  f  (x)  =  B  +  2  Ox  +  3BX2  +  ^Exz  +  &c. 

f"  (x)  =  2O  +  3  .  2Bx  +  4 .  3EX2  +  &o. 
f"(x)  =  3  .  2B  +  4 .  3  .  2Ex  +  &c. 

Hence,  making  x  =  o  in  each  of  these  equations,  we  get 
/(o)  =  A,    /'(o)  =  5,    €M  =  C,  •  g^  =  D,  &o. 

whence  we  obtain  the  same  series  as  before. 

The  preceding  expansion  is  usually  called  Maclaurin's* 
Theorem ;  it  was,  however,  previously  given  by  Stirling,  and 
is,  as  is  shown  already,  but  a  particular  case  of  Taylor's  series. 
We  proceed  to  illustrate  it  by  a  few  examples. 

60.  Exponential  Series. — Let  y  =  ax. 

Here        f(x)    =  ax,  hence  f(o)    =1, 

f(x)    =axloga,  „     f(o)  =  loga, 

f(x)  =a*(loga)\  „     /»=log«)2, 

/(»)  (x)  =  a*  (log  a)\  „     /W  (o)  =  (log  a) n  ; 

and  the  expansion  is 

(xloga)      (xloga)2  (x  log  a)n      n  ,     . 


/»  _ 


1  .  2  .  . .  n 


If  6,  the  base  of  the  Napierian  system  of  Logarithms,  be 
substituted  for  a,  the  preceding  expansion  becomes 

e1 =  1  +  -  + +  ...+ +  ...  (12) 

11. 2  1  . 2  ...  ra 


*  Maclaurin  laid  no  claim  to  the  theorem  which  is  known  by  his  name,  for, 
after  proving  it,  he  adds — "This  theorem  was  given  by  Dr.  Taylor,  Method. 
Increm."     See  Maclaurin's  Fluxions,  vol.  ii.,  Art.  751. 

F 


66  Development  of  Functions. 

If  x  =  i  this  gives  for  e  the  same  value  as  that  adopted  in 
Art.  29,  viz.  : 

111  1 

e  =  1  +  — h + + +  .  .  . 

1       1.2       1.2.3      1.2.3.4 

61.  Expansion  of  sin  x  and  cos  x  by  Maclaurin's 
Theorem.     Let /(a?)  =  sin  a?,  then 

/(o)=o,    /(o)-i,    /»  =  o,    /"(o)  =  -  I,  &c, 

and  we  get 

sm  x  = + &c 

1      1.2.3      1.2.3.4.5 


In  like  manner 


COS  X  =   I + 


1.2.3.4 


the  same  expansions  as  already  arrived  at  in  Art.  58. 

Since  sin  (-  x)  =  -  sin  a?,  we  might  have  inferred  at  once 
that  the  expansion  for  sin  x  in  terms  of  x  can  only  consist  of 
odd  powers  of  x.  Similarly,  as  cos  (-  x)  =  cos  x,  the  expan- 
sion of  cos  x  can  only  contain  even  powers. 

In  general,  if  F(x)  =  F(-  x),  the  development  of  F(x) 
can  only  consist  of  even  powers  of  x.  If  F(—  x)  =  -  F(x)t  the 
expansion  can  contain  odd  powers  of  x  only. 

Thus,  the  expansions  of  tana?,  sin-1a?,  tan-1#,  &c,  can  con- 
tain no  even  powers  of  x ;  those  of  cos  x,  sec  x,  &c,  no  odd 
powers. 

62.  Iff ny  gens'  Approximation  to  length  of  Circular 
Arc.* — If  A  be  the  chord  of  any  circular  arc,  and  B  that  of 

half  the  arc ;  then  the  length  of  the  arc  is  equal  to ,  q.p. 

■J 

For,  let  R  be  the  radius  of  the  circle,  and  L  the  length  of 
the  arc  :  and  we  have 

A         .LB         .     L 

R  =  2mise  B  =  2Bm^ 

— — - — i 

*  This  important  approximation  is  due  to  Huygens.  The  demonstration 
given  above  is  that  of  Newton,  and  is  introduced  by  him  as  an  application  of 
his  expansion  for  the  sine  of  an  angle.     Vid.  "  Epis.  Prior  ad  Oldemburgium." 


Huygens'  Approximation.  67 

hence,  by  (8), 

X3  L5 

A  =  L =r9  + 7T-  -  &C. 

2.3.  4.  B?     2  .3  .4.  5  .  16.  i24 

T3  7"  5 

8J5  =  4Z — -  + -  -  &o. 

2  .  3  .  4  .  i£2      2  . 3  .  4 .  5  .  64  .  B* 

consequently,  neglecting  powers  of  —  beyond  the  fourth,  we 
get 

sb -a  _/      x4  \  •; 

Hence,  for  an  arc  equal  in  length  to  the  radius  the  error  in 

adopting  Huygens'  approximation  in  less  than         th  part  of 

the  whole  arc ;  for  an  arc  of  half  the  length  of  the  radius 
the  proportionate  error  is  one-sixteenth  less ;  and  so  on. 
In  practice  the  approximation*  is  used  in  the  form 

L  =  2B  +  -  (2B-A). 
3 

This  simple  mode  of  finding  approximately  the  length  of 
an  arc  of  a  circle  is  much  employed  in  practice.  It  may  also 
be  applied  to  find  the  approximate  length  of  a  portion  of 
any  continuous  curve,  by  dividing  it  into  an  even  number  of 
suitable  intervals,  and  regarding  the  intervals  as  approxi- 
mately circular.  See  Eankine's  Rules  and  Tables,  Part  I., 
Section  4. 

*  To  show  the  accuracy  of  this  approximation,  let  us  apply  it  to  find  the 
length  of  an  arc  of  300  in  a  circle  whose  radius  is  100,000  feet. 

Here  B  =  2R  sin  70  30',     A  =  iR  sin  150 ; 

hut,  from  the  Tahles, 

sin  70  30'  =  .1305268,     sin  150  =  .2588190. 

„  „      2B-A 

Hence  2JB  + =  523^9.71. 

The  true  value,  assuming  ir  =  3.1415926,  is  52359.88  ;  whence  the  error  is  but 
.17  of  a  foot,  or  about  2  inches. 

F  2 


68  Development  of  Functions. 

63.  Expansion  of  tan_1#. — Assume,  according  to  Art. 
61,  the  expansion  of  tan-1a?to  be 

Ax  +  Bxz  +  Cx5  +  Bx1  +  &c, 

where  A,  B,  C,  &c,  are  undetermined  coefficients : 

then  — '—= =  A  +  $Bx2  +  5O4  +  7D06  +  &c. ; 

ax 

1    .               01  *  tan  x         1  o       a       n      o 

but  = = =  i  -  x2,  +  or  -  xs  +  &c, 

ax  1  +  xz 

when  x  lies  between  the  limits  ±  1 . 
Comparing  coefficients,  we  have 

A  =  i,    B  =  --,     C  =  \    D  =  --,  &c. 

3  5  7 

Hence 

/Vi  /ViO  /y,D  /yi&lVT  1 

tan-1#  = + ...  +  (-  i)n +  .  .  . ;      (14) 

135  v      '   2n  +  1  v     ' 

when  x  is  less  than  unity. 

This  expansion  can  be  also  deduced  directly  from  Mac- 
laurin's  Theorem,  by  aid  of  the  results  given  in  Art.  46. 
This  is  left  as  an  exercise  for  the  student. 

64.  .Expansion  of  sin_1#. — Assume,  as  before, 

sin-1#  =  Ax  +  Bx3  +  Cx5  +  &c. ; 
then  7 ^  =  A  +  sBxz  +  sCx*  -^  &c. ; 

(1  -  x~y 

but  ,     Vxl  =  (1  -  ^2)-*  =  1  +  -x*  +  —  04  +  .  .  . 

(1  -  a?)  4      v  '  2  2.4 


1  .  3  .  .  .  2r  -  1    _ 

+ #2r  +  .  .  . 

2.4...     2r 


Hence,  comparing  coefficients,  we  get 


Finally, 


A=i,    B=-.-,     C  =  ±-^-.-,&c. 

23  2.45 


.    ,       x     ix3     1  .  3   x5  1 . 3 . . .  2r  -  1     arrH  .     x 

sin_1a?  =  -  +  -  .-  +  — -.  —  +  .. .  +  — . +...  (15) 

1     23     2.45  2.4...    2r      2r+  1 


Eider's  Expressions  for  Sine  and  Cosine.  69 

Since  we  have  assumed  that  sin-1a?  vanishes  along  with  x  we 
must  in  this  expansion  regard  sin"1^  as  being  the  circular 
measure  of  the  acute  angle  whose  sine  is  x. 

There  is  no  difficulty  in  determining  the  general  formula 
for  other  values  of  sm~lx,  if  requisite. 

A  direct  proof  of  the  preceding  result  can  be  deduced 
from  Maclaurin's  expansion  by  aid  of  Art.  5 1 .  We  leave 
this  as  an  exercise  for  the  student. 

From  the  preceding  expansion  the  value  of  ir  can  be 
exhibited  in  the  following  series : 

7T      1        111.31. 

-  =  -  + + +  &c. 

6      2      2.38      2. 4. 532 

I  7T  I 

For,  since  sin  30°  =  -,  we  have  -  =  sin"1-  ;  .'.  &c. 

2  62 

An  approximate*  value  of  it  can  be  arrived  at  by  the  aid 
of  this  formula ;  at  the  same  time  it  may  be  observed  that 
many  other  expansions  are  better  adapted  for  this  purpose. 

65.  Enler's  Expressions  for  Sine  and  Cosine. — In 

the  exponential  series  (12),  iix*/  -  1  be  substituted  for  x, 
we  get 

/yi2  nA 

gXV-i  _  j +   . , +  &0.   .  .  . 

1.2         I.2.3.4 


+ 


^ 


+  &G 


I         1.2.3 

=  cos  x  +  v  -  1  sin  x ;  by  Art.  59. 

Similarly,  e~x,/~l  =  cos  x  -  */  -  1  sin  x. 
Hence  ex^~x  +  e"^"1  =  2  cos  x,  ) 


o*"/-l     ._  />-%>/ 


~1  =  2A/- 


(16) 


-  #■**->■  =  2a/  -  i  sin  x. 


A  more  complete  development  of  these  formulae  will  be 
found  in  treatises  on  Algebra  and  Trigonometry. 

*  The  expansion  for  snr1^,  and  also  this  method  of  approximating  to  ir,  were 
given  by  Newton. 


7°  Development  of  Functions. 

66.  John  Bernoulli's  Series. — If,  in  Taylor's  Ex- 
pansion (i)  we  make  y  =  -  x,  and  transfer  fix)  to  the  other 
side  of  the  equation,  we  get 


x 


f(x)  =/(o)  +  xf(x)  -  ~  /"(co)  +  -^—  f"(x)  -  &o.     (17) 

1.2  1.2.3 

This  is  equivalent  to  the  series  known  as  Bernoulli's,* 
and  published  by  him  in  Act.  Lips.,  1694. 

As  an  example  of  this  expansion,  let/  (x)  =  ex  ;  then 

/(o)  =  1,    fix)  =  ex,    f"(x)  =  e\  &c, 
and  we  get 

ex  =  1  +  xex ex  +  &c, 

1 .  2 

Or,  dividing  by  ex,  and  transposing, 

a? 

6*  =    I  -  X  + &0.f 

\'2 

which  agrees  with  Art.  60. 

67.  Symbolic   Form   of  Taylor's   Theorem. — The 

expansion 

may  be  written  in  the  form 

/(„„,=  j,.„|+i(|)V...|(|)V..J/w,(,s) 

in  which  the  student  will  perceive  that  the  terms  within  the 
brackets  proceed  according  to  the  law  of  the  exponential 
series  (12)  ;  the  equation  may  accordingly  be  written  in  the 
shape 

f{x  +  y)  =  eyrxf(x),  (19) 

*  In  his  Heduc.  Quad,  ad  long,  curv.,  John  Bernoulli  introduces  this  theorem 
again,  adding — "  Quam  eandum  seriem  postea  Taylorus,  interjecto  viginti 
annorum  intervallo,  in  librum  quern  edidit,  a.d.  1715,  demethodo  incrementorum, 
transferre  dignatus  est  sub  alio  tanturn  characterum  habitu."  The  great  in- 
justice of  this  statement  need  not  be  insisted  on ;  for  while  Taylor's  Theorem  is 
one  of  the  most  important  in  the  entire  range  of  analysis,  that  of  Bernoulli  is 
comparatively  of  little  use ;  and  is,  as  shown  above,  but  a  simple  case  of  Taylor's 
Expansion. 


Symbolic  Form  of  Taylor's  Theorem.  7 1 

d 

where  e  *x  is  supposed  to  be  expanded  as  in  the  exponential 

theorem,  and V^  written  for  f-  (  —  )  fix),  &c. 

\n     dxn  \n  \dxj  J  x  ' 

This  form  of  Taylor's  Theorem  is  of  extensive  application 
in  the  Calculus  of  Finite  Differences. 

68.  Other  Forms  derived  from  Taylor's  Series. — 

In  the  expansion  (3),  Art.  55,  substitute  h  for  y, 

,,  hdu        h2    d2u  hn        dnu     0 

then    u±  =  u  +  -  —  + —  +  .  .  . ;—  +  &c. 

1  dx      1.2  dx2  1  .  2  ...  n  axn 

If  now  h  be  diminished  indefinitely,  it  may  be  represented 
by  dx,  and  the  series  becomes 

du  dx     d2u  dx2  dnu      dxn 

u,  =  u  +  - +  -r-   —  +  .  .  .  +  — - 


dx  1       dx2  1  .  2  dxn  1  .  2  .  .  .  n 

or         u,-u=  "^  dx  +  J^-  dx2  +  £^&-  dx"  +  &c,      (20) 
1  1.2  1-2.3 

in  which  ux  -  u  is  the  complete  increment  of  ^,  corresponding 
to  the  increment  dx  in  #. 

Again,  since  each  term  in  this  expansion  is  infinitely  small 
in  comparison  with  the  preceding  one,  if  all  the  terms  after 
the  first  be  neglected  (by  Art.  38)  as  being  infinitely  small  in 
comparison  with  it,  we  get 

du  =ff(x)  dx, 

the  same  result  as  given  in  Art.  7. 

Another  form  of  the  preceding  expansion  is 

du       d2u  d3u  dnu  0  .     . 

Ui  -  U  =  —  + I h  .  .  .  H +  &C.  (21) 

1        1.2       1.2.3  i  .  2  .  . .  n  ' 

69.  Theorem. — If  a  function  of  x  become  infinite  for  any 
finite  value  of  x  then  all  its  successive  derived  functions  become 
infinite  at  the  same  time. 

If  the  function  be  algebraic,  the  only  way  that  it  can  be- 
come infinite  for  a  finite  value  of  x  is  by  its  containing  a 

P 

term  of  the  form  -=,  in  which   Q  vanishes  for  one  or  more 


72  Development  of  Functions. 

values  of  x  for  which  P  remains  finite.     Accordingly,  let 

dPPdQ 

_  P    .       clu  _  dx      Q  dx ;  this  also  becomes  infinite  when 

Q '  dx  Q 

Q  =  o. 

ft  if  fJ  11 
Similarly,  -r-0,  — - ,  &c,  each  become  infinite  when  Q  =  o. 
dx1  dxz 


\ 


Again,  certain  transcendental  functions,  such  as  e  , 
cosec  (x  -  a),  &c,  become  infinite  when  x  =  a;  but  it  can  be 
easily  shown,  by  differentiation,  that  their  derived  functions 
also  become  infinite  at  the  same  time.  Similar  remarks  apply 
in  all  other  cases. 

The  student  who  desires  a  more  general  investigation  is 
referred  to  De  Morgan's  Calculus,  page  179. 

70.  Remarks  on  Taylor's  Expansion. — In  the  pre- 
ceding applications  of  Taylor's  Theorem,  the  series  arrived 
at  (Art.  56  excepted)  each  consisted  of  an  infinite  number  of 
terms ;  and  it  has  been  assumed  in  our  investigation  that  the 
sum  of  these  infinite  series  has,  in  each  case,  &  finite  limiting 
value,  represented  by  the  original  function, /(#  +  y),  or  fix). 
In  other  words,  we  have  assumed  that  the  remainder  of  the 
series  after  n  terms,  in  each  case,  becomes  infinitely  small 
when  n  is  taken  sufficiently  large — or,  that  the  series  is  con- 
vergent. The  meaning  of  this  term  will  be  explained  in  the 
next  Article. 

71.  Convergent  and  ^Divergent  Series. — A  series, 
Mi,  u2,  u3,  .  .  .  un,  .  .  .  consisting  of  an  indefinite  number  of 
terms,  which  succeed  each  other  according  to  some  fixed  law, 
is  said  to  be  convergent,  when  the  sum  of  its  first  n  terms 
approaches  nearer  and  nearer  to  a  finite  limiting  value,  accord- 
ing as  n  is  taken  greater  and  greater ;  and  this  limiting  value 
is  called  the  sum  of  the  series,  from  which  it  can  be  made  to 
differ  by  an  amount  less  than  any  assigned  quantity,  on 
taking  a  sufficient  number  of  terms.  It  is  evident  that  in  the 
case  of  a  convergent  series  the  terms  become  indefinitely 
small  when  n  is  taken  indefinitely  great. 

If  the  sum  of  the  first  n  terms  approximates  to  no  finite 
limit  the  series  is  said  to  be  divergent. 


Convergent  and  Divergent  Series.  73 

In  general,  a  series  consisting  of  real  and  positive  terms 
is  convergent  whenever  the  snm  of  its  first  n  terms  does  not 
increase  indefinitely  with  n.  For,  if  this  sum  do  not  become 
indefinitely  great  as  n  increases,  it  cannot  be  greater  than  a 
certain  finite  value,  to  which  it  constantly  approaches  as  n 
is  increased  indefinitely. 

72.  Application  to  Geometrical  Progression. — 
The  preceding  statements  will  be  best  understood  by  apply- 
ing them  to  the  case  of  the  ordinary  progression 

I  +  X  +  x2  +  xs  +  .  .  .  +  xn  +  .  .  . 

I  —  xn 

The  sum  of  the  first  n  terms  of  this  series  is in  all  cases. 

1  -  x 

(1).  Let#<  1 ;  then  the  terms  become  smaller  and  smaller 
as  n  increases ;  and  if  n  be  taken  sufficiently  great  the  value 
of  xn  can  be  made  as  small  as  we  please. 

Hence,  the  sum  of  the  first  n  terms  tends  to  the  limiting 

value ;  also  the  remainder  after  n  terms  is  represented 

1     x 


fln 


X 

by ,  which  becomes  smaller  and  smaller  as  n  increases, 

I  - —  x 

and  may  be  regarded  as  vanishing  ultimately. 

(2).  Let  x  >  1.     The  series  is  in  this  case  an  increasing 
one,  and  xn  becomes  infinitely  great  along  with  n.     Hence 

T     _    /V>W  /Ylft    T 

the  sum  of  n  terms, or ,  as  well  as  the  remainder 

1 -x         x -  1 

after  n  terms,  becomes  infinite  along  with  n.     Accordingly 

the  statement  that  the  limit  of  the  sum  of  the  series 

1  +  x  +  x2  +  .  .  .  +  xn  +  .  .  .  ad  infinitum 

is holds  only  when  x  is  less  than  unity,  i.  e.  when  the 

1     x 

series  is  a  convergent  one. 

In  like  manner  the  sum  of  n  terms  of  the  series 

I  -  X  +  X*  -  X3  +  &c. 

I   -(-   l)nXn 

is — . 

I   +  X 


74  Development  of  Functions. 

As  before,  when  x  <  i,  the  limit  of  the  sum  is ;    but 

i  +  x 

when  x  >  i,  xn  becomes  infinitely  great  along  with  n,  and  the 
limit  of  the  sum  of  an  even  number  of  terms  is  -  co  ;  while 
that  of  an  odd  number  is  +  oo  .  Hence  the  series  in  this  case 
has  no  limit. 

73.  Theorem. — If,   in  a  series  of  positive  terms  repre- 
sented by 

Ui  +  u2  +  .  .  .  +  un  +  &c, 

the  ratio  ~^—  be  less  than  a  certain  limit  smaller  than  unity,  for 
un 

all  values  of  n  beyond  a  certain  number ',  the  series  is  convergent, 

and  has  a  finite  limit. 

Suppose  k  to  be  a  fraction  less  than  unity,  and  greater 

than  the  greatest  of  the  ratios  -^  .  .  .  (beyond  the  number 

un 

n),  then  we  have 

<  A/,  .*.  Ufi+i  <  HUfi' 

^w+2         Z-  •  TA 

*C  it,  .  .  UinYl  *"*  ">  lA/xi. 

Mn+l 
Un+r 


<~  K,  .  .  Ufi+r  "^-  it  Ufi' 


hi+r-i 


Hence,  the  limit  of  the  remainder  of  the  series  after  un  is 
less  than  the  sum  of  the  series 

kun  +  k2un  +  .  .  .  +  krun  ...        ad  infinitum  ; 

therefore,  by  Art.  72,  less  than 

- ; ,  since  k  <  1 . 

1  -k 

Hence,  since  un  decreases  as  n  increases,  and  becomes  infi- 
nitely small  ultimately,  the  remainder  after  n  terms  becomes 
also  infinitely  small  when  n  is  taken  sufficiently  great ;  and 
consequently,  the  series  is  convergent,  and  has  a  finite  limit. 

Again,  if  the  ratio  ~^-  be  >  1,  for  all  values  of  n  beyond 

un 


Convergent  and  Divergent  Series.  75 

a  certain  number,  the  series  is  divergent,  and  has  no  finite 
limit.  This  can  be  established  by  a  similar  process;  for, 
assuming  k  >   1,   and  less  than  the  least   of  the  fractions 

-^-,  .  .  .  then  by  Art.  72  the  series 

un  +  kun  +  khtn  +  &c.  ad  infinitum 
has  an  infinite  value ;  but  each  term  of  the  series 

Un  +  %i+i  +  Un+2  +  &C. 

is  greater  than  the  corresponding  term  in  the  above  geome- 
trical progression ;  hence,  its  sum  must  be  also  infinite,  &c. 
These  results  hold  also  if  the  terms  of  the  series  be  alter- 
nately positive  and  negative ;  for  in  this  case  k  becomes 
negative,  and  the  series  will  be  convergent  or  divergent 
according  as  -  k  is  <  or  >  1 ;  as  can  be  readily  seen. 

In  order  to  apply  the  preceding  principles  to  Taylor's 
Theorem  it  will  be  necessary  to  determine  a  general  expres- 
sion for  the  remainder  after  n  terms  in  that  expansion ;  in 
order  to  do  so,  we  commence  with  the  following : — 

74.  liemiiaa. — If  a  continuous  function  (f>(x)  vanish  when 
x  =  a,  and  also  when  x  =  b,  then  its  derived  function  §\x\  if 
also  continuous,  must  vanish  for  some  value  of  x  between  a 
and  b. 

Suppose  b  greater  than  a;  then  if  <p'{x)  do  not  vanish 
between  a  and  b,  it  must  be  either  always  positive  or  always 
negative  for  all  values  of  x  between  these  limits;  and 
consequently,  by  Art.  6,  (p(x)  must  constantly  increase,  or 
constantly  diminish,  as  x  increases  from  a  to  b,  which  is 
impossible,  since  (p(x)  vanishes  for  both  limits.  Accordingly, 
(j/(x)  cannot  be  either  always  positive  or  always  negative  ; 
and  hence  it  must  change  its  sign  between  the  limits,  and, 
being  a  continuous  function,  it  must  vanish  for  some  inter- 
mediate value. 

This  result  admits  of  being  illustrated  from  geometry. 
For,  let  y  =  0  (x)  represent  a  continuous  curve ;  then,  since 
<p  (a)  =  o,  and  (j>  (b)  =  o,  we  have  y  =  o,  when  x  =  a,  and  also 
when  x  =  b  ;  therefore  the  curve  cuts  the  axis  of  x  at  distances 
a  and  b  from  the  origin  ;    and  accordingly  at  some  inter- 

I 


76  Development  of  Functions. 

mediate  point  it  must  have  its  tangent  parallel  to  the  axis  of 
x.  Hence,  hy  Art.  10,  we  must  have  $f(x)  =  o  for  some 
value  of  x  between  a  and  b. 

75.  Lagrange's  Theorem  on  the  ILimits  of  Tay- 
lor's Series. — Suppose  Rn  to  represent  the  remainder  after 
n  terms  in  Taylor's  expansion,  then  writing  X  for  x  +  y  in  (1), 
we  shall  have 

+  ^^P /<"-'>  (*)  +  p»,  (22) 

in  which  f(x),  f'{x) /(n)  (a?)  are  supposed  finite  and 

continuous  for  all  values  of  the  variable  between  X  and  x. 

From  the  form  of  the  terms  included  in  Bn  it  evidently 
may  be  written  in  the  shape 

(x  -  xy 

\n 

where  P  is  some  function  of  X  and  x. 
Consequently  we  have 


+ 


— 1 — -  -P I  =  o.  (23) 


Now,  let  g  be  substituted  for  x  in  every  term  in  the  pre- 
ceding, with  the  exception  of  P,  and  let  F(%)  represent  the 
resulting  expression  :  we  shall  have 

F{z)  =/(X)  -  [/(,)  +  {*^f  («)  +  ...  +  ^^"  P  j,   (24) 

in  which  P  has  the  same  value  as  before. 

Again,  the  right-hand  side  in  this  equation  vanishes 
when  2  =  X;  .:  F(X)  =  o. 

Also,  from  (23),  the  right-hand  side  vanishes  when  z=x; 
.-.  P(^)  =  o. 

J 


Limits  of  Taylor's  Series.  7  7 

Accordingly,  since  the  function  F  (z)  vanishes  when  z  =  X9 
and  also  when  z  =  x9  it  follows  from  Art.  74  that  its  derived 
function  Ff(z)  also  vanishes  for  some  value  of  z  between  the 
limits  X  and  x. 

Proceeding  to  obtain  F\z)  by  differentiation  from  equa- 
tion (24),  it  can  be  easily  seen  that  the  terms  destroy  each 
other  in  pairs,  with  the  exception  of  the  two  last.  Thus  we 
shall  have 


W   -   I  IW 


Consequently,  for  some  value  of  z  between  x  and  X  we 
must  have 

/W  («)  =  P. 

Again,  if  0  be  a  positive  quantity  less  than  unity  it  is 
easily  seen  that  the  expression 

x  +  0  (X  -  x), 

by  assigning  a  suitable  value  to  0,  can  be  made  equal  to  any 
number  intermediate  between  x  and  X. 
Hence,  finally, 

P=/W  {0+0  (X-a>)}, 

where  0  is  some  quantity  >  o  and  <  1 . 

Consequently,  the  remainder  after  n  terms  of  Taylor's 
series  can  be  represented  by 

A  =  £j^>>  {»  +  «(*-*)}.  (as) 

Making  this  substitution,  the  equation  (22)  becomes 

/(X)  =/(x)  +  £^V  (*)  +  ^f^-V  W  +  . .  • 

"  "'  '  -/<•*)  W  +^f-^/W  {*  +  0  (X-a)}.        (26) 


+ 


w  —  1  \n 


The  preceding  demonstration  is  taken,  with  some  slight 
modifications,  from  Bertrand's  "  Traite  de  Calcul  Differentiel" 
(273). 


78  Development  of  Functions. 

Again,  if  h  be  substituted  for  X  -  x,  the  series  becomes 
f(x  +  h)  =ffa)  +hf{x)  +&c. 

h71-1    „ hn 


+ 


f{n-i)  ^  +         f(n)  (^  +  0/^#  (27) 


»  —  I  W 


In  this  expression  n  may  be  any  positive  integer. 
If  n  =  1  the  result  becomes 

fix  +  h)  =f(x)  -r  hf  (X  +  Oh).  (28) 

When  n  =  2, 

/(»  +  A)  -/(«.)  +  hf  (0)  +  -£-/"  (0  +  0/*).  (29) 

The  student  should  observe  that  0  has  in  general  different 
values  in  each  of  these  functions,  but  that  they  are  all  subject 
to  the  same  condition,  viz.,  0  >  o  and  <  1. 

It  will  be  a  useful  exercise  on  the  preceding  method  for 
the  student  to  investigate  the  formulse  (28)  and  (29)  inde- 
pendently, by  aid  of  the  Lemma  of  Art.  74. 

The  preceding  investigation  may  be  regarded  as  furnish- 
ing a  complete  and  rigorous  proof  of  Taylor's  Theorem,  and 
formula  (27)  as  representing  its  most  general  expression. 

76.  Geometrical  Illustration. — The  equation 

f(X)  =f(x)  +  (X-x)f  {x+0(X-x)} 

admits  of  a  simple  geometrical  verification;  for,  let  y  =ffa) 
represent  a  curve  referred  to  rectangular  axes,  and  suppose 
(X,  Y),  (x,  y)  to  be  two  points  Px,  P2  on  it :  then 

f(X)-f(x)      r-y 

X  -  x  X  -  x' 

Y  -  v 
But  = — -  is  the  tangent  of  the  angle  which  the  chord  Pi  P2 

■A     —  X 

makes  with  the  axis  of  x ;  also,  since  the  curve  cuts  the 
chord  in  the  points  Pi,  P2,  it  is  obvious  that,  when  the  point  on 
the  curve  and  the  direction  of  the  tangent  alter  continuously, 
the  tangent  to  the  curve  at  some  point  between  Px  and  P2  must  be 
parallel  to  the  chord  Px  P2 ;  but  by  Art.  10,  f  fa)  is  the  tri- 
gonometrical tangent  of  the  angle  which  the  tangent  at  the 


Second  Form  of  Remainder. 


79 


point  (a>i,  yx)  makes  with  the  axis  of  x.    Hence,  for  some  value, 
xlf  between  X  and  x,  we  must  have 

ff,   y-y  /m-/w 

;  W"X-«"      x-*     ' 

or,  writing  xx  in  the  form  a?  +  9  (X  -  x), 

/(X)  =/(*)  +  (X  -  x)f  [x  +  9  (X-x)}. 

77.  Second  Form   of  Remainder. — The  remainder 
after  n  terms  in  Taylor's  Series  may  also  be  written  in  the  form 


W—  I 


For  it  is  evident  that  Rn  may  be  written  in  the  form 
(X  -x)P1; 

.-.AX) =/{x) +(x-x)  f\x) + . . . + ^rT"1/^  w 


w  -  I 


+  (X  -  0)  Pl 

Substitute  g  for  x,  as  before,  in  every  term  except  P1 ;  and  the 
same  reasoning  is  applicable,  word  for  word,  as  that  employed 
in  Art.  75.  The  value  of  Ff  (z)  becomes,  however,  in  this 
case 


n  -  1 


and,  as  F'(z)  must  vanish  for  some  value  of  %  between  x  and 
X,  we  must  have,  representing  that  value  by  x  +  9  (X  -  x), 


=   (X        X)^    (l         9Y-1  f[n)    ^  +  q  (x  _  xy^ 


n 


(30) 


where  9,  as  before,  is  >  o  and  <  1 . 

If  h  be  introduced  instead  of  X-x,  the  preceding  result 
becomes 


Rn  =  il S!!1  #»/(»)  (x  +  Oh), 


n-  1 


(31) 


which  is  of  the  required  form. 


80  Development  of  Functions. 

Hence,  Taylor's  Theorem  admits  of  being  written  in  the 
form 

/(.+  *)  =/ (0)  +  \f\x)  +  -*!_/»+...  +  -^/M  (•) 


n—  i 


■  -^—  (1  -  O^fW  (x  +  Oh).        (32) 
w  - 1 


The  same  remarks  are  applicable  to  this  form*  as  were  made 
with  respect  to  (27). 

From  these  f  ormulse  we  see  that  the  essential  conditions 
for  the  application  of  Taylor's  Theorem  to  the  expansion  of 
any  function  in  a  series  consisting  of  an  infinite  number  of 
terms  are,  that  none  of  its  derived  functions  shall  become 
infinite,  and  that  the  quantity 

£/«(*  + 0A) 

shall  become  infinitely  small,  when  n  is  taken  sufficiently 
large ;  as  otherwise  the  series  does  not  admit  of  a  finite  limit. 

78.  Limit  of when  n  is  indefinitely  great. 

1  .  2  ,,n 

Let  un  = ,  then  —  = ;  .*.  —  becomes  smaller 

1 .  2  . .  n  un      n  +  1  un 

and  smaller  as  n  increases ;  hence,  when  n  is  taken  sufficiently 
great,  the  series  uMl,  un+2,  .  .  .  &c,  diminishes  rapidly,  and 
the  terms  become  ultimately  infinitely  small.  Consequently, 
whenever  the  nth  derived  function  fW  (x)  continues  to  be  finite  for 
all  values  of  n,  however  great,  the  remainder  after  n  terms  in 
Taylor's  Expansion  becomes  infinitely  small,  and  the  series  has 
a  finite  limit. 


*  This  second  form  is  in  some  cases  more  advantageous  than  that  in  (27). 
An  example  of  this  will  be  found  in  Art.  83. 


Remainder  in  the  Expansion  of  sin  x.  8 1 

79.    General   Form    of  Maclanrin's    Series. — The 

expansion  (27)  becomes,  on  making  x  =  o,  and  substituting 
x  afterwards  instead  of  h, 

/(•)  -/(o)  +7/(0)  +-^-2/"(°)  +  •  •  •  +  i|zi/(M)  (°) 


+  -/W(fe).  (33) 

Hence  the  remainder  after  n  terms  is  represented  by 

-m  (Ox)  ; 

where  6  is  >  o  and  <  1. 

This  remainder  becomes  infinitely  small  for  any  function 

xn 
f(x)  whenever  r-f^  (Ox)  becomes  evanescent  for  infinitely 

\n 

great  values  of  n. 

We  shall  now  proceed  to  examine  the  remainders  in  the 
different  elementary  expansions  which  were  given  in  the 
commencement  of  this  chapter. 

80.  Remainder  in  the  Expansion  of  ax. — Our  for- 
mula gives  for  Rn  in  this  case 

~  CLosaYa9*. 

\n 

Now,  aQx  is  finite,  being  less  than  ax ;  and  it  has  been  proved 

( x  I02*  a\n 
in  Art.  78  that  - — s        becomes  infinitely  small  for  large 

values  of  n.  Hence  the  remainder  in  this  case  becomes 
evanescent  when  n  is  taken  sufficiently  large.  Accordingly 
the  series  is  a  convergent  one,  and  the  expansion  by  Taylor's 
Theorem  is  always  applicable. 

8 1 .  Remainder  in  the  Expansion  of  sin  x. — In  this 

case 

D       xn    .    frnr      Q  \ 
Rn=  \ —  sm    —  +  ux  . 

\n         \  2 

G 


82  Development  of  Functions. 

This  value  of  Rn  ultimately  vanishes  by  Art.  78,  and  the 
series  is  accordingly  convergent. 

The   same  remarks   apply  to  the  expansion   of   cos   x. 
Accordingly,  both  of  these  series  hold  for  all  values  of  x. 

82.  Remainder  in  the  Expansion  of  log  (1  +  x). — 
The  series 

/y»  /ys-*  /ViO  /yi* 

+ +  &c, 

1234 

when  x  is  >  1,  is  no  longer  convergent ;  for  the  ratio  of  any 
term  to  the  preceding  one  tends  to  the  limit  -  x ;  conse- 
quently the  terms  form  an  increasing  series,  and  become 
ultimately  infinitely  great.  Hence  the  expansion  is  inappli- 
cable in  this  case. 

1.2.      n 1^ 

Again,  since  fn(x)  =  (-  iV*-1     *     *  * '  x- — -,  the  remainder 

6      '  J    v  '     v      ;  (1  +  x)n 

Rn  is  denoted  by  - — —  ( tt-  )  |;  hence,  if  x  be  positive  and 

x 
less  than  unity,  — —-r-  is  a  proper  fraction,  and  the  value  of 

I  ~r  UX 

Rn  evidently  tends  to  become  infinitely  small  for  large  values 
of  n  ;  accordingly  the  series  is  convergent,  and  the  expansion 
holds  in  this  case. 

83.  Binomial  Theorem  for  Fractional  and  Nega- 
tive Indices. — In  the  expansion 

m        m  (m  -  1)   _ 

(  I  +  x)m ■  =  I  +  —  X  +  — -x2  +  .  .  .  . 

I  1.2 

•m(m-  1)  .  .  .  (m-n+  i)xn      0 

+  — * - —  +  &c. 

1  .  2  .  . .  n 

if  un  denote  the  nth  term,  we  have 

un+1     m-n+  1 


x, 


un  n 


the  value  of  which,  when  n  increases  indefinitely,  tends  to 
become  -  x ;  the  series,  accordingly,  is  convergent  if  x  <  1, 
but  is  not  convergent  if  x  >  1 . 


Binomial  Theorem.  83 

Accordingly,  the  Binomial  Expansion  does  not  hold  when 
x  is  greater  than  unity. 
Again,  as 

/(»)  (x)  =  m  (m  -  1)  .  .  .  (m  -  n  +  1)  (1  +  #)m_w, 

the  remainder,  by  formula  (25),  is 

m  (m-  1)  .  .  .  (m-n  +  1)      '.        n  .m_in 
i  .  2  .  .  .  n 
or 

m  (m-  1)  .  .  .  (m-n  +  1)         xn 


1  .  2  .  .  .  n  (1  +  6x)n-m' 

Now,  suppose  x  positive  and  less  than  unity ;  then,  when 
n  is  very  great,  the  expression 

m(m—  1)  .  .  .  (m-n+  1) 
i  .  2  .  .  .  n 

becomes  indefinitely  small ;  also m  is  less  than  unity; 

(  I    "T   UX) 

hence,  the  expansion  by  the  Binomial  Theorem  holds  in  this 
case. 

Again,  suppose  x  negative  and  less  than  unity.  We  employ 
the  form  for  the  remainder  given  in  Art.  77,  which  becomes 
in  this  case 

.     .   m(m-  1)  .  .  .  (m-n+  i)xn  ,        n7„"T/       n  \™  n 

(-  i)n— \ r — '- —  (1  -  O)"-1  (1  -  Qx)m~n  ; 

'  1  . 2  .  .  .  [n-  i)  '      x  ' 


or 


.     .m(m-  1)  .  .  .  (m-  n+  1)  (1  -  0)m~1afi 
'_I)  1.2  ...  (n-  1) 


-0 


X 


1  —  0 

Also,  since  x<  1,  Qx<  0;  .*.  1  -  Ox  >  1  -  0  ;  hence pr- 

1  -  ux 

is  a  proper  fraction ;  .*.  any  integral  power  of  it  is  less  than 

unity ;  hence,  by  the  preceding,  the  remainder,  when  n  is 

sufficiently  great,  tends  ultimately  to  vanish. 

g  2 


84  Development  of  Functions, 

In  general  (x  +  y)m  may  be  written  in  either  of  the  forms 
Wi+^j   orywfi+- 

now,  if  the  index  m  be  fractional  or  negative,  and  x  >  y,  or 

y 

-  a  proper  fraction,  the  Binomial  Expansion  holds  for  the 

x 

series 

(x  +  y)m  =  xml  1  +  ^ )  =  #"»  +  —  oj^y  +  — ^ xm'2y2  +  &c, 

but  does  not  hold  for  the  series 

/        v  /       x\m  m       ,       m(m-i) 

(x+y)m  =  ymli  +-J  =ym  +  -  ym~1x  +  — K- *  ^"2^2  +  &c, 

since  the  former  series  is  convergent  and  the  latter  divergent. 

We  conclude  that  in  all  cases  one  or  other  of  the  expan- 
sions of  the  Binomial  series  holds ;  but  never  both,  except 
when  m  is  a  positive  integer,  in  which  case  the  number  of 
terms  is  finite. 

84.  Remainder  in  the  Expansion  of  tan"1^. — The 
series 

/yi  /ytO  /yiD 

.  -  1*/  w  i/U  * 

tan-1#  = + &c, 

1      3       5 

is  evidently  convergent  or  divergent,  according  as  x  <  or  >  1 . 
To  find  an  expression  for  the  remainder  when  x<  1,  we  have, 

ky  (8)>  p.  50— 


/wW-(s)-tlHrt"-(-1)** 


\n  -  1  .  sin  [n n  tan~!# 


(i  +  x*)% 
Hence  we  have,  in  this  case, 


#TOsin  hi n  tan~ 


1  m  J 


W(l  +  0V)S  ' 

which,  when  x  lies  between  +  1  and  -  1,  evidently  becomes 
infinitely  small  as  n  increases,  and  accordingly  the  series  holds 
for  such  values  of  x. 


Expansion  by  aid  of  Differential  Equations.  85 

85.  Expansion  of  sin"1  a?. — Since  the  function  sin"1  a?  is 
impossible  unless  x  be  <  1,  it  is  easily  seen  that  the  Series 
given  in  Art.  64  is  always  convergent ;  for  its  terms  are  each 
less  than  the  corresponding  terms  in  the  geometrical  pro- 
gression 

x  +  xz  +  x5  +  &c. 

Consequently,  the  limit  of  the  series  is  always  less  than  the 
limit  of  the  preceding  progression. 

A  similar  mode  of  demonstration  is  applicable  to  the 
expansion  of  tan-1  x  when  x  <  1,  as  well  as  to  other  analogous 
series. 

In  every  case,  the  value  of  Bni  the  remainder  after  n 
terms,  furnishes  us  with  the  degree  of  approximation  in  the 
evaluation  of  an  expansion  on  taking  its  first  n  terms  for 
its  value. 

86.  Expansion  by  aid  of  Differential  Equations. — 
In  many  cases  we  are  enabled  to  find  the  relation  between 
the  coefficients  in  the  expansion  of  a  function  of  x  by  aid  of 
differential*  equations ;  and  thus  to  find  the  form  of  the 
series. 

For  example,  let  y  =  ex,  then 

dy 

—  =  ^  =  y 

dx  y* 

Now  suppose  that  we  have 

y  =  a0  +  axx  +  a2x2  +  .  .  .  anxn  +  .  .  . , 

then  ~  =  ax  +  2d%x  +  .  .  .  nanxn~l  +  &c. 

dx 

Accordingly  we  have 

«i  +  2azx  +  3%#3  +  .  .  .  =  a0  +  aix  +  azx2  +  &c, 


*  This  method  is  indicated  by  Newton,  and  there  can  be  little  doubt  that  it 
was  by  aid  of  it  he  arrived  at  the  expansion  of  sin  (m  sin-1  x),  as  well  as  other 
series. — Vide  Ep.  posterior  ad  Oldemburgium.  It  is  worthy  of  observation  that 
Newton's  letters  to  Oldemburg  were  written  for  the  purpose  of  transmission  to 
Leibnitz. 


86  Development  of  Functions. 

hence,  equating  coefficients,  we  have 

#1  =  CC0,      Cl%—  —  =  — ,       Clz  =  —  =  ,  &C. 

2  2  3         2.3 

Moreover,  if  we  make  x  -  o,  we  get  a0  =  1, 

.'.  ex  =  i  +  -  + +  +  &c, 

I         1.2         I.2.3 

the  same  series  as  before. 
Again,  let 

y  =  sin  (m  sin-1  x) . 

Here,  by  Art.  47,  we  have 

d2y        dy 

Now,  if  we  suppose  y  developed  in  the  form 

y  =  a0  +  axx  +  a2x2  +  .  .  .  +  anxn  +  &c, 

du 
then  —  =  ax  +  2a2x  +  xazx2  +  .  .  .  +  nanxn~x  +  &c, 

ax 

d2y 

-zr-  =  2a2  +  3  .  2a%x  +  .  .  .  +  n(n  -  i)anxn~2  +  &c. 

dx2  x  7 

Substituting  and  equating  the  coefficients  of  xn  we  get 

ri~  -  m2  , 

(n  +  1)  (■«  +2) 

Again,  when  0  =  o  we  have  y  =  o ;  .*.  #0  =  o. 

Hence  we  see  that  the  series  consists  only  of  odd  powers 
of  x  ;  a  result  which  might  have  been  anticipated  from  Art. 
61. 

To  find  ax.    When  x  =  o,  cos  (m  sin-1^)  =  1 ,  hence  l—\=m; 

accordingly  ax  =  m  ; 

m2  -  1  m  (m2  -  1 ) 

• .  ci%  =  — #1  =  « 

2.3  1.2.3 

m2  -  9  m  (m8  -  1 )  (w2  -  9)  m 

&§   =   —    #3    = • 

4-5  1.2.3.4.5 


Expansion  of  sin  mz  and  cos  mz.  87 

hence  we  get 

•  *  /      •    ,  \      m       m  (m2  ~  -1)  •, 
sin*  (m  sm"^   =  —  x xs 

1  1.2.3 

+  mK-i).K-9)^5_&c>  (     } 

I.2.3-4-5 

In  the  preceding,  we  have  assumed  that  sin-1^  is  an  acute 
angle,  as  otherwise  both  it,  and  also  sin  (m  siirt?),  would  admit 
of  an  indefinite  number  of  values. — See  Art.  26. 

87.  Expansion  of  sin  mz  and  cos  mz. — If,  in  (35),  2  be 
substituted  for  sin-1#,  the  formula  becomes 

( 1       m2  -  1     . 

sin  mz  =  m  sin  z  I suns 

(1      1.2.3 

+  (nf-  1)  (m»  -  9)  sin%  .  &)  (6) 

In  a  similar  manner  it  can  be  proved  that 

m2  sin2s      m2  (m2  -  4)    .   .        D  .     . 

cos  mz  =  1 +    — surs  -  &c.  (37) 

1.2         1.2.3.4 

If  m  be  an  odd  integer  the  expansion  for  sin  mz  consists 
of  a  finite  number  of  terms,  while  that  for  cos  mz  contains  an 
infinite  number.  If  m  be  an  even  integer  the  number  of 
terms  in  the  series  for  cos  mz  is  finite,  while  that  in  sin  mz  is 
infinite. 

The  preceding  series  hold  equally  when  m  is  a  fraction. 

A  more  complete  exposition  of  these  important  expansions 
will  be  found  in  Bertrand's  "  Calcul  Differentiel." 

In  general,  in  the  expansion  (36),  the  ratio  of  any  term 

to  that  which  precedes  it  is  -, 7—, r  sin2s,  which,  when 

x  (n  +  1)  (n  +  2) 

n  is  very  great,  approaches  to  suns.     Hence,  since  sin  z  is 

less  than  unity,  the  series  is  convergent  in  all  cases.     Similar 

observations  apply  to  expansion  (37). 

*  This  expansion  is  erroneously  attributed  to  Euler  by  M.  Bertrand ;  it  was 
originally  given  by  Newton.     See  preceding  note. 


88  Development  of  Functions. 

The  expansion 


.    ,  ax     a2x2      a  {a2  +  i2)    _     a2  (a*  +  22) 

easm-ix  =I+  —  +  +  _^ L  xz  +  _^ £  ar  +  .  .  . 

I  1.2  1.2.3  I.2.3.4 

can  "be  easily  arrived  at  by  a  similar  process. 

88.  Arbogast's  Method  of  derivations. 

X  X2  x^ 

If  u  =  a  +  b-  +  c +  d +  &c, 

1         1.2         1.2.3 

to  find  the  coefficients  in  the  expansion  of  <p  (u)  in  ascending 
powers  of  x — 

Let  f{x)  =  $  (u)9 

B  C 

and  suppose  f(x)  =  A  +  —  x  + x2  +  &c. 

=/(o)+f/'(o)+-^I/"(o)+&o.) 

then  we  have  evidently 

A=f{6)  =0(fl). 
Also,  writing  u',  u",  u"',  &c.  instead  of 
du      d2u      d3u    . 

by  successive  differentiation  of  the  equation /(a?)  =  $  (u),  we 
obtain 

f  (x)  =  <j>'  (u)  .  u, 

f\x)  =  0'  {u)  .  n    +  <p"  (u)  .  (it')2, 

f"(x)  =  $  (u)  . «/"  +  3<t>"  M  •  ^  •  ^  +  0'"  M  (O3, 

f*(x)  =  $  (u)  .  &  +  <j>"  (u)  O'  u'"  +  3  KO2]  +  6f'M  •  M*-  **" 

+  0iv  (w)  .  (V)4. 

Now,  when  x  =  o,  w,  w',  ?/',  w'",  .  .  .  obviously  become 
a,  b,  c,  d,  .  .  .  respectively. 


Arbogasfs  Method  of  Derivations.  89 

Accordingly, 

B  =  f  (o)  =  0'  (a)  .  ft, 

C  =  /"  (o)  =  0'  (a)  .  0  +  f  (a)  .  J», 

D  =  /'"  (o)  =  0'  (a) .  rf  +  3tf>"  (a) .  fc  +  0'"  {a) .  b\ 

E  =  /*  (o)  =  0'  (a)  .  e  4  tf>"  (of)  (4-bd  +  3c2)  +  60'"  [a) .  &2c 

+  0iv  (a) .  bK 

From  the  mode  of  formation  of  these  terms,  they  are  seen 
to  be  each  deduced  from  the  preceding  one  by  an  analogous 
law  by  that  to  which  the  derived  functions  are  deduced  one 
from  the  other  ;  and,  as /'(#),/"(#)  .  .  .  are  deduced  from/(a?) 
by  successive  differentiation,  so  in  like  manner,  B,  C,  D,  .  .  . 
are  deduced  from  0  (u)  by  successive  derivation  ;  where,  after 
differentiation,  a,  b,  c,  &c,  are  substituted  for 

du      dhc  „ 

'  dx      dx2J 

If  this  process  of  derivation  be  denoted  by  the  letter  8,  then 

B  =  3.A,     C=$.B,    D  =  8.<7,  &c.  (38) 

From  the  preceding,  we  see  that  in  forming  the  term 
8  .  0(#<),  we  take  the  derived  function  0'(#),  and  multiply  it 
by  the  next  letter  b,  and  similarly  in  other  cases. 

Thus     8  .  6    =  c,  8  .  c  =  d,  .  .  . 

8  .  bm  =  mbm~'c,        8  .  cm  =  mcm-\l  .  .  . 
Also  8  .  0'  (a)  b  =  <p'(a)c  +  0"  (a)  b\ 

This  gives  the  same  value  for  C  as  that  found  before  ;  D 
is  derived  from  C  in  accordance  with  the  same  law ;  and  so 
on. 

The  preceding  method  is  due  to  Arbogast :  for  its  com- 
plete discussion  the  student  is  referred  to  his  "  Calcul  des 
Derivations."  The  Rules  there  arrived  at  for  forming  the 
successive  coefficients  in  the  simplest  manner  are  given  in 
"  Gralbraith's  Algebra,"  page  342. 


90  Development  of  Functions. 

As  an  illustration  of  this  method,  we  shall  apply  it  to  find 
a  few  terms  in  the  expansion  of 

sm  [a  +  o  -  +  c +  d +  &c. 

\  i         1.2         1.2.3 

Here  A  =  sin  a,    B  =  8  .  sin  #  =  6  cos  #, 

C  =  $  .  b  cos  «  =  c  cos  #  -  52  sin  0, 

J)  =  g  .  C  =  d  cos  <z  -  3^c  sin  a  -  b%  cos  a, 

J£  =  <$  .  D  =  e  oosa  -  (4^  +  3c2)  sin  a  -  6b2c  cos  a 

+  ¥  sin  #. 

If  the  series  a  +  bx  +  c +  &c.  consist  of  a  finite  num- 

1  .  2 

ber  of  terms  the  derivative  of  the  last  letter  is  zero — thus,  if 

d  be  the  last  letter,  8  .  d  =  o,  and  d  is  regarded  as  a  constant 

with  respect  to  the  symbol  of  derivation  S. 

If  the  expansion  of  $  (u)  be  required  when  u  is  of  the 

form 

a  +  fix  +  yxz  +  $x3  +  &c, 

the  result  can  be  attained  from  the  preceding  method  by 
substituting  a,  b,  c,  d,  &o.  instead  of  a,  ]3,  1  .  2  y,  1  .  2  .  3  .  §, 
&c,  and  proceeding  as  before. 

The  student  will  observe  that  in  the  expression  for  the 
terms  D,  E,  &c,  the  coefficients  of  the  derived  functions 
$'(a),  <j/(a)9  &c,  are  completely  independent  of  the  form  of  the 
function  <f>,  and  are  expressed  in  terms  of  the  letters,  b,  c,  d, 
&c.  solely ;  so  that,  if  calculated  once  for  all,  they  can  be  applied 
to  the  determination  of  the  coefficients  in  every  particular 
case,  by  finding  the  different  derived  functions  $'  (a),  <j>"(a), 
&c,  for  that  case,  and  multiplying  by  the  respective  coef- 
ficients, determined  as  stated  above. 


Examples.  g  i 


Examples. 

i.  If  it  =  f(ax+bu),  then  -  —  =  t  ^r-     This  furnishes  the  condition  that 
J  v  ^"  adx      b  dy 

a  given  function  of  x  and  y  shou  d  be  a  function  of  ax  +  by. 

2.  Find,  by  Maclaurin's  theorem,  the  first  three  terms  in  the  expansion  of 
tan  x. 

X3         2X5 

Ans.  x  + 1 . 

3        i5 


3.  Find  the  first  four  terms  in  the  expansion  of  sec  x. 

xz      c#4      61  x6 
Ans.   1  +  —  +  —  +  . 

2  24        720 

4.  Find,  by  Maclaurin's  theorem,  as  far  as  #4,  the  expansion  of  log  (1  +  sin  x) 
in  ascending  powers  of  x. 

Let  f(x)  =  log  (1  +  sin  x), 

,,  cos  a;         1  -sino; 

then/  (x)  = : —  = =  sec  x  —  tan  x, 

i  +  sma;         cos  x 

f"(x)  =  sec  x  tan  x  —  sec2x  =  -  f'(x)  sec  x  ; 
.'.  f'"{x)  =  —f"(x)  sec  x  —f'(x)  sec  a;  tana;, 

/iv  (x)  —  _/"'(#)  sec  x  -  2f"(x)  sec  a;  tan  x  -f'(x)  (2  sec3#  -  sec  x) ; 
.'./(o)  =  o,    /'(o)  =  l,    /"(o)=-i,    /'"(<>)  =  1,    /iv(o)  =  -2; 

yy»2  /y»3  />*4 

.\  log  (1  +  sin  x)  =  # +  -7 1-  &c. 

0  N  '  2612 

ex 

5.  Find  six  terms  of  the  development  of  ■ in  ascending  powers  of  x 

cos  X 

„        2#3        #4        3a;5 

Ans.  1  +  x  +  x2  -i 1 —  +  —  .  •  . 

3  2        10 

6.  Apply  the  method  of  Art.  86,  to  find  the  expansions  of  sin  x  and  cos  x. 

7.  Prove  that 

i  /       r\      j.               »    •      sins      .,    .     .„  sin  2s       ._    .         sin  32; 
tan"1  (x  +  h)  =  tan"i  x  +  h  sin  z (h  sin  zf V  (h  sin  z)3 cec, 

where  a  =  cot-1*. 

if  dnz 

Here/(a;)  =  tan-1i»  =  —  z;  and  by  Art.  46,  —  =  (-  i)»J^-i  sinwzsin«z;  .\&c. 


92  Examples. 

8.  Hence  prove  the  expansion 

ir  sin  z  sin  2z       „      sin  3s 

-  =  2  +   COS  2  H COS2Z  + C0S3Z  +  &C. 

2123 

Let  h  —  —  cot  z  —  —  x,  &c. 

g.  Prove  that 

ir      z      sin  2      sin  22      sin  3z 

-  =  -  + + + +  &c. 

2      2         1  2  3 

Let  h  sin  z  =  -  1  in  Example  7  :  then  h  +  x  —  — : =  -  tan  - ;  .  \  &c. 

sms  2 

10.  Prove  the  expansion 


it      sin  z      1  sin  2z      1  sin  iz 

-  =  ■ + 5-  +  -  — ~  +  &C. 

2         COS  Z        2    COS4  2  3  C0Sd2 

Assume  h  — ,  then 

sin  2  cos  z 

#  +  h  =  -  tan2  =  tan  (ir  -  z) ;  .:  w  —  z  ■=  tan-1  (#  +  A),  &c. 

Suhstituting  in  Example  7 ,  we  get  the  result  required. 
The  preceding  expansions  were  first  given  by  Euler. 

11.  Prove  the  equations 

sin  9%  =  9  sin  x  —  120  sin3#  +  432  sin5 a:  -  576  sin'ta  +  256  sin9a;, 
cos  6x  =  32  cos6#  —  48  cos4#  +  18  cos2#  —  1. 

These  follow  from  the  formulse  of  Article  78. 

12.  If  m  =  2,  Newton's  formula,  Art.  87,  gives 

(  .  sin3#      sin%       .     ) 

sin  2X  =  2    sins; &c.  >  ; 

(  2         2.4  ) 

verify  this  result  by  aid  of  the  elementary  equation  sin  2%  =  2  sin  x  cos  x. 

13.  If  <p  (x  +  h)  +  (p  (x  —  h)  =  <p  (x)  (p  (h),  for  all  values  of  x  and  h, 

(b"(x)      4>iv(v) 
prove  that  ■    ,  .    =  z-^r-r  =  &c.  =  constant ; 

<p(x)       <p"(x) 

and  also  <f>'(o)  =  o,        <}>'"(o)  =  o,  &c. 

14.  If,  in  the  last,  ^-7-/  =  a%'>  prove  that  <p{x)  =  eax  +  ^ax, 

<p{x) 

If  — j-~-  =  -a2;  prove  that  <p  (x)  =  2  cos  (##). 


Examples.  93 

15.  Apply  Arbogast's  method  to  find  the  first  four  terms  in  the  expansion 
of 

{a  +  bx  +  ex2  +  dx3  +  &c.)M. 

(iyi  lift  J^  \ 

— b%  +  nac  J  a11'2  x2 
1.2  J 

+  n  |(*~  ^^"2A^-s  b3  +  (n-i)  or*  be  +  a^d\  x*  +  &c. 

&"  +  1 

16.  Prove  that  the  expansion  of .  x  can  contain  no  odd  powers  of  a?. 

For  if  the  sign  of  x  he  changed,  the  function  remains  unaltered. 

17.  Hence,  show  that  the  expansion  of contains  no  odd  powers  of  x 

beyond  the  first. 

_                                        x         x     x    ex  +  1  . 

Here  +  -  =  -  . ;  .*.  &c. 

ex  —  1      2      2    ex  —  1 

18.  If  u  = ,  prove  that 

ex  -  1 

w  (dn-xu\        n(n-i)  /dn~2u\  (du\        ,  . 

1  U^)„  +  -T7-  U^)„  +  '  •  •  +n  Ujo+  (M)0  =  o; 

and  hence  calculate  the  coefficients  of  the  first  five  terms  in  the  expansion  of  u* 
Here  exu  =  x  +  u,  and  by  Art.  48,  we  have 

(du      n{n—i)dHi  dnu\       dnu 

dx         1.2     dx2      '  '  '     dxn)      dxn' 

x  x        B\  Bz  .  B%  , 

IQ.     If  =  I  -  -  + -  X2 - X*  + 2  x    ~  '  •  • 

ex  —  1  2      1.2  1.2.3.4  1. 2. ..6 


prove  that 


B\  =  -?,     Bi  —  — ,      -B3  =  — ,      Bi  =  — ,  &c. 
6  3°  4*  3° 


These  are  called  Bernoulli's  numbers,  and  are  of  importance  in  connexion 
with  the  expansion  of  a  large  number  of  functions. 

20.  Prove  that 

x         x      Bi%2  B2x*        .  ■  BZ2* 

(22  _  1)  + (24  -  1) (2«  -  1)  +  .  . 


ex  +  1      2      1.2  1.2.3.4  1  .  2  ...  6 


94  Examples. 

21.  Hence,  prove  that 

^Z_L=fe(22_l)  +  ^(24_I)+^fL^(2«-l)+&C. 

e*+i  v  y       3-4  3-4-5.6 

a;      #3        a;5       „ 

= + &c. 

2      24      240 

22.  Prove  that 

22^i«2  24_B2£4  26^3^ 

a;  cot  x  =  1 7  -  ^c- 

1.2        1.2.3.4      1. 2... 6 

X  2?2%P 

23.  Also,  tan  -  =  Bxx  (22  -  1)  +  - —  (2*  -  1)  +  &c. 

2  3-4 

24.  Prove  that 

x        x  x*  at      B$x* 

-  cot  -  =  1  —  2?i -. Bit- 77 

22  |2  |4         |6 

This  follows  immediately  by  substituting  -  for  x  in  Ex.  22. 

25.  Given  u  (u  —  x)  =  1 ;  find  the  four  first  terms  in  the  expansion  of  u  in 
terms  of  x,  by  Maclaurin's  Theorem. 

d2i/      dy 

26- If  *dS+dx+y=o> 

expand  y  in  powers  of  x  by  the  method  of  indeterminate  coefficients. 

27.  Show  that  the  series 

x       x2      xz       #4 

jt»        2m        Zm         Am 

is  convergent  when  x  <  1,  and  divergent  when  x  >  1,  for  all  values  of  m. 

28.  Prove  the  expansion 

f(x)  =        I       /(ft)  1  ^    (/(«)) 

(#  -  a)m  £  (a?)       (#  -  «)m  <?>(«)      (a;  -  a)w-1  ^#  (^(«) J 

,    * MV(^)+&c 

I  .  2  .  (aj  -  a)™-*  \da]    \<p{a)) 

29.  Find,  by  Maclaurin's  Theorem,  the  first  four  terms  in  the  expansion  of 

(1  +  x)x  in  ascending  powers  of  x. 

1 
Let  f(x)  =  (1  +  x)x, 


Examples.  95 

then  /'(*)  _/(.)  (_!_  -  l?lO_tf}\ 

/'"(*)  =  -/"(*)  (^  -  ^  +  \&  -&o.)  +  a/»  (^  -  f  *  +  &o.J  . 
But,  by  Art.  29,  /(o)  =  «; 

.»  &£      ilea;2      7^    o 

Hence  (i+x)x  =  e h -^kH  &c. 

v         '  2        24         16 

This  result  can  be  verified  by  direct  development,  as  follows : 

1 

let  u  =  (1  +  #)x, 

T  SC         OS  W 

then  log  u  =  -  log  (1  +  x)  =  1  —  + 1- .  .  . : 

a?  234 

*       2       3  a;2     xZ 

.-.  u  =  e   2    T    *"'  '*  =  *>.  e  2   ~*     4  '  " 

r       (x     x2     x*     \     x2  (i     x    x2     \~      x3    (i     x         \3 
LI_\2_I+4"/2  \2~3  +  4    7  ~2TiU"  3+     /    ""J 

[a?      ii#2      7a;3 
I--  + i-7  .... 
2        24         16  J 

30.  In  Art.  76,  if  /(as)  and/'(a:)  be  not  both  continuous  between  the  points 
Pi,  P2,  show  that  there  is  not  necessarily  a  tangent  between  those  points,  parallel 
to  the  chord. 

31.  Find  the  development  of - in  ascending  powers  of  as,  the  coef- 

sin  x  sin  2x 

ficients  being  expressed  in  Bernoullian  numbers.     "  Camb.  Math.  Trip.,  1878." 

_.  x  sin  "\x 

Since : =  x  cot  x  +  x  cot  2x,  the  expansion  in  question,  by  (22), 

sin  x  sin  2x  '  r  ^  >    *  \     '» 

is 

3       22^2^2,  ,       24j54^4   „  ,         N      26B6x*  e  = 

I T7-(2+i) rf- (2'  +  l)-— nr-  (25  +  i)-  &c. 


(     96     ) 


CHAPTEE  IY. 


INDETERMINATE   FORMS. 


89.  Indeterminate  Forms. — Algebraic  expressions  some- 
times become  indeterminate  for  particular  values  of  the 
variable  on  which  they  depend ;  thus,  if  the  same  value  a 
when  substituted  for  x  makes  both  the  numerator  and  the 

f  (x)  f  (a) 

denominator  of  the  fraction  —7—;  vanish,  then  ~~-  becomes  of 

<t>(x)  (p{a) 

the  form  -,  and  its  value  is  said  to  be  indeterminate. 
o 

Similarly,  the  fraction  becomes  indeterminate  if  /  (x)  and 
<t>  (x)  both  become  infinite  for  a  particular  value  of  x.  We 
proceed  to  show  how  its  true  value  is  to  be  found  in  such 
cases.  By  its  true  value  we  mean  the  limiting  value  which  the 
fraction  assumes  when  x  differs  by  an  infinitely  small  amount 
from  the  particular  value  which  renders  the  expression  indeter- 
minate. 

It  will  be  observed  that  the  determination  of  the  diffe- 
rential coefficient  of  any  expression/^)  may  be  regarded  as  a 
case  of  finding  an  indeterminate  form,  for  it  reduces  to  the 

determination  of  — '- — — —  when  h  =  o. 

h 

In  many  cases  the  true  values  of  indeterminate  forms  can 
be  best  found  by  ordinary  algebraical  and  trigonometrical 
processes. 

We  shall  illustrate  this  statement  by  a  few  examples. 

Examples. 

m,     »       .      ax2  -  2acx  +  ac2  .  „;1     .        O  ,        . 

1.  The  fraction 5  becomes  of  the  form  -  when  x  =  o ;  but  since 

bx2  —  2bcx  +  oci  o 

it  can  be  written  in  the  shape  — 7 ^.  its  true  value  in  all  cases  is  -. 

r    b{x  -c)2  b 


Examples.  97 

x  o 

2.  The  fraction  — z=== .  becomes  -  when  x  —  o. 

a/ a  +  x  -a/  a  —  x  ° 

To  find  its  true  value,  multiply  its  numerator  and  denominator  by  the  com- 
plementary surd,  a/  a  +  x  +  a/  a  —  %,  and  the  fraction  becomes 

x\a/ a  +  x  +  a/  a  —  x)       a/  a  +  x  +  a/  a  —  x 
—LL r i  or : 


the  true  value  of  which  is  a/  a  when  x  =  o. 

-  V  a2  +  ax  +  x2  —  \/a2  —  ax  +  x2 

i-  — — — ,  when  x  =  o. 

a/  a  +  x  —  a/  a  —  x 
Multiply  by  the  two  complementary  surd  forms,  and  the  fraction  becomes 

2ax  {a/  a  +  x  +  a/ a  -  x} 
2x  {a/  a^  +  ax  +  x2  +  a/ a2— ax+x2} 

a  [a/  a  +  x  4-  v  a  —  x) 
a/  a2  +  ax  -J-  x2  +  a/  a2  —  ax  +  a2 

the  true  value  of  which  evidently  is  a/  a  when  x  =  o.  From  the  preceding 
examples  we  infer  that  when  an  expression  of  a  surd  form  becomes  indeter- 
minate, its  true  value  can  usually  be  determined  by  multiplying  by  the  com- 
plementary surd  form  or  forms. 


or 


2X  —  a/  §x2  —  a2      ,  1 

4.        when  x  —  a.  Am.  -. 

x  —  V2a;2  —  a2  2 

a  -  a/  a2  —  x2     ,  1 

5. when  x  =  o.  Am.  — . 

xz  2a 


,  a  sm  0  -  sin  a9  .  o     . 

o.         -- 1 — -  becomes  -  when  0  =  o. 

0(cos  0  -  cos  ad)  o 

To  find  its  true  value,  substitute  their  expansions  for  the  sines  and  cosines, 
and  the  fraction  becomes 

/  03  \  /    n  «303  \ 

a    0 +  .  .  •)  -  lad +  ... 

V        1-2.3  /        V         1.2.3  / 

„ '      e2  a2e2 

e{ +  ...+ .. .} 

1    1 . 2  1.2        J 

03 

g-  (a3  -  a)  +  . . . 


or 


03 


H 


98  Indeterminate  Forms. 

Divide  by  03  (a2  -  1),  and  since  all  the  terms  after  the  first  in  the  new  numerator 

and  denominator  vanish  when  0  =  o,  the  true  value  of  the  fraction  is  -  in  this 

3 
case. 


7.  The  fraction 

A0xm  +  Aiccm~l  +  A2xm~2  +  .  .  .  At 
a0xn  +  aixn~l  +  ...+«» 


becomes  ~  when  x  =  00 : 


its  true  value  can,  however,  be  easily  determined,  for  it  is  evidently  equal  to 
that  of 

A\      A% 

A0  +  —  +  —  +  . .  . 
x        x^ 


%  +  -  +  -5 + 

X         X6 


Moreover,  when  x  =  00,  the  fractions  — ,  —  ...—  ...,  all  vanish :    hence, 

x     x*         x 

the  true  value  of  the  given  fraction  is  that  of 

A0    , 
%m-n  —  when  x  —  00. 

The  value  of  this  expression  depends  on  the  sign  of  m  —  n. 

(i.)  If  m  >  n,  xm'n  =  00  -when  x  =  00 ;  or  the  fraction  is  infinite  in  this 
case. 

An 

(2.)  If  m  =  n,  the  true  value  is  — . 

a0 

(3.)  If  m  <  n,  then  xm-n  =  o  when  x  =  00  ;  and  the  true  value  of  the  frac- 
tion is  zero. 

Accordingly,  the  proposed  expression,  when  x  =  00,  is  infinite,  finite,  or  zero, 
according  as  m  is  greater  than,  equal  to,  or  less  than  n.     Compare  Art.  39. 

8.  u  =  *y  x  +  a  -  y/ x  +  b,  when  x  =  00. 

__  a  —  b  . 

Here  u  =  =  o  when  x  =  00. 

■\/x  +  a  +  \/  x  +  b 

9.  -v/^3  +  ax  —  x>  when  x  =  00.  -4ws.  -. 

10.  w  =  «*  sin  f  —  K  when  a;  =  00. 


(?) 


(1.)  If  a  <  i,  ax  —  o  when  a;  =  00,  and  therefore  the  true  value  of  u  is  zero 
in  this  case. 

c   .    . 
(2.)  If  a  >  I,  then  a*  becomes  infinite  along  with  #  ;  but  as  —  is  infinitely 

Or 
G  C 

small  at  the  same  time,  we  have  sin  —  =  — .     Hence,  the  true  value 

ax     ax 

of  u  is  e  in  this  case. 


Method  of  the  Differential  Calculus.  99 

1 1.  u  =  a/  a%  —  xl  cot  -  A is  of  the  form  oxm  when  x  =  a. 

2  \  a  +  x 

Here 

but,  when  a  -  x  is  infinitely  small, 

la  —  x      it     la 
v  a  +  x      2  \  a 


+ 

*y  a2  —  x2 

.a  -  x 
tan 

a  +  x 


IT 

tan 


2  >/  a  +  x      2  v  a  +  x 


\/  a?  —  x'1      a  +  x      4a 

u  =        ,        -  = =  —  when  x  =  a. 

-  x         *  t 


ir    la 
l\a 


+  x 


x  sin  sin  #)  —  sin2#      , 
12.  w  = r^ ,  when  x  =  o. 

Substitute  the  ordinary  expansion  for  sin  x,  neglecting  powers  beyond  the  sixth, 
and  u  becomes 


(  .  sin3a;      sin5x)        /        #3      x5\ 

x  I  sin  x — -  +  - —  [  -   lx-r  +  —  ) 

(  3  5    )       V         3       5/ 


xz      x5 \  2 

-  -4-    i    —     i  v.  — 

|3 

X% 

xz      Xs      1  /        #3  \ 3     #5  /  a;3      xi  \ 2 

|3      |5     6\        J3  /       \5  \         |3      |5/ 

x6 

Hence  we  get,  on  dividing  by  x5,  the  true  value  of  the  fraction  to  be  —  when 

I  o 
X  =  O. 

(asin^  +  gcos^)*-^  . 

13-  ■ an-pn '  When  a  =  &-  ■Ans-   Sm $> 

Similar  processes  may  be  applied  to  other  cases ;  there 
are,  however,  many  indeterminate  forms  in  which  such  pro- 
cesses would  either  fail  altogether,  or  else  be  very  laborious. 

We  now  proceed  to  show  how  the  Differential  Calculus 
furnishes  us  with  a  general  method  for  evaluating  indetermi- 
nate forms. 

90. — Method  of  the  Differential   Calculus. — Sup- 

f(x)  .  o 

pose  -—-!-  to  be  a  fraction  which  becomes  of  the  form  -  when 
(j>{x)  o 

x  =  a; 

i.  e.f(a)  =  o,  and  0  (a)  =  o ; 

h  2 


ioo  Indeterminate  Forms. 

substitute  a  +  h  for  x  and  the  fraction  becomes 

f(a  +  h)  -f(a) 

f(a  +  h)  h 

or 


(p(a  +  h)y        (j)(a  +  h)  -  ${a)  * 
~         h 

but  when  h  is  infinitely  small  the  numerator  and  denominator 
in  this  expression  become  f'(a)  and  (j>'(a)9  respectively;  hence, 
in  this  case, 


f(a  +  h)     f(a) 

(p(a  +  h)      <p'(a)' 


Accordingly,  —jtA  represents  the  limiting  or  true  value  of 
the  fraction 


(i.)  If /'(a)  =  o,  and  $ (a)  be  not  zero,  the  true  value  of 

f{a). 

— t-£  is  zero. 

(2.)  If  /'(#)  be  not  zero,  and  <p'(a)  =  o,  the  true  value  of 

(3.)  If  ff{a)  =  o,  and  (f>'(a)  =  o,  our  new  fraction  -}t4  *s 

still  of  the  indeterminate  form  -.     Applying  the 
preceding  process  of  reasoning  to  it,  it  follows  that 

ftr(ri\ 

its  true  value  is  that  of    „,  !. 

<j>  (a) 

If  this  fraction  be  also  of  the  form  -,  we  proceed  to  the 

next  derived  functions. 

In  general,  if  the  first  derived  functions  which  do  not 

vanish  he  fW(a)  and  #(*%)>  then  the  true  value  of  " 


isthatof^ 


4>W 


Examples,  101 

Examples. 


if 
x  sm  x 

i.  u=  ,  when  x  =  -. 

COS  #  2 


Here  /(a;)  =  x  sin  # , 


<p  (x)  =  cos  a; ; 
■'•  f\x)  =  #  cos  a;  +  sin  #,  /'  f  -  J  =  i, 

<f)'(x)  =  —  sin x,  <p'  I  -  j  =  -  i. 

Hence  u  =  —  i,  when  #  =  -. 

2 

2.  w  =  — — ■,  when  x  =  a. 

(x  -  a)r 

Here  /(#)  =  0™*  -  ema, 

tp{x)  =  (x  -  a)r  ; 

.'.  f'(x)  =  me™*,  f'(a)  =  mema. 

<l>'(x)  =  r(x  —  ay-1,  <l>\a)  is  o  or  oo,  as  r  >  or  <  i. 

Hence  the  true  value  of  u  is  oo  or  o,  according  as  r  >  or  <  i . 

This  result  can  also  be  arrived  at  by  writing  the  fraction  in  the  form 

S  gm(x-a) j  "J.  gma       gmh j 


(x  -  a)r  hr 


ema,  where  h  =  x  —  a 


hence,  expanding  emh,  and  making  h  =  o,  we  evidently  get  the  same  result  as 
before. 


x  —  sin  x 


3- 

— - —  w 

nen  a;  =  o. 

Here 

f'(x)  —  I  -  cos#, 

/'(o)=o 

$'(x)  =  sA 

<j>'(o)  =  o. 

f"(x)  =  sin  a;, 

/"(o)=o. 

(j>"(x)  =  6x, 

*»  =  o. 

f'"(x)  =  COS#, 

/'"(o)  =  i. 

<p'"(x)  =  6, 

4>'"(o)  =  6. 

io2  Indeterminate  Forms. 


Hence,  the  true  value  is  7,  as  can  also  be  immediately  arrived  at  by  substituting 
6 

#3 
x  —  —  +  &c.  instead  of  sin  x. 
o 

4.  when  x  =  o.  Ans.  log  a. 

x 

e*f(x)-e°f(a)^_^  /(«)  +/'(a) 

5*  r~; 7~x  when  #  =  #.  ,,  — — tttt. 

«*$>(#)  -  ea(j)(a)  $(a)+  tf{a) 

It  may  be  obseived  that  each  of  these  examples  can  be  exhibited  in  the  form 
00  —  00  ,  that  is,  as  the  difference  of  two  functions  each  of  which  becomes  in- 
finite for  the  particular  value  of  x  in  question. 

91.  Form  o  x  00. — The  expression/^)  x  ty(x)  becomes 
indeterminate  for  any  value  of  x  which  makes  one  of  its  fac- 
tors zero  and  the  other  infinite.     The  function  in  this  case  is 

easily  reducible  to  the  form  -  ;  for  suppose  f{a)  =  o,  and  $  (a) 

■Pi  \ 
=  00,  then  the  expression  can  be  written  ,  which  is  of  the 

required  form. 

Examples. 

ttX 

1.  Find  the  value  of  (1  —  x)  tan  —  when  x  =  i. 

This  expression  becomes  .  the  true  value  of  which  is  -  when  x  —  r. 

r  irx  7t 

cot  — 
2 


I  x  sin  x j , 


2.      Sec  x  (  x  sin  x  -     ) ,  when  x  =  -. 


IT 

x  sin  x  — 
2 
This  becomes  .  a  form  already  discussed, 

cos  x 

3.  Tan  (x  -  a)  .  log  (x  -  a),  when  x  =  a.  Ans.  o. 


4.  Cosec2/3^ .  log  (cos  «as),  „    #  =  0. 


a- 
»>  —  Tp3* 


2/32 


Method  of  the  Differential  Calculus.  103 


92.  Form  ^-.     As  stated  before,  the  fraction  -^  also 


becomes  indeterminate  for  the  value  x  =  a,  if 
/(#)  =  00,  and  0  (a)  =  00. 

It  can,  however,  be  reduced  to  the  form  -  by  writing  it 
in  the  shape 


0(g) 
1 


A") 

The  true  value  of  the  latter  fraction,  by  Art.  90,  is  that  of 

fix) 
Now,  suppose  .4  represents  the  limiting  value  of  -j-\ 

(p  (X) 

when  x  =  a,  then  we  have 

f\d)  <f>'(a) 

that  is,  the  true  value  of  the  indeterminate  form  ^  is  found 

in  the  same  manner  as  that  of  the  form  -. 

o 

In  the  preceding  demonstration,  in  dividing  both  sides  of 

our  equation  by  A9  we  have  assumed  that  A  is  neither  zero 

nor  infinity  ;  so  that  the  proof  would  fail  in  either  of  these 

cases. 

It  can,  however,  be  completed  as  follows  : — 

■pi  \ 
Suppose  the  real  limit  of  -W  to  be  zero,  then  that  of 

0(«) 

.  >         is  k,  where  k  may  be  any  constant ;  but  as  the 

<p(a) 


104  Indeterminate  Forms. 

latter  fraction  has  a  finite  limit,  its  value  by  the  preceding 
method  is 

1*     -     /'(a)        .      1      ,,  •         /»  •    1 

therefore     ,;  ;  =  o ;  i.e.  when  -4  is  zero,  —rA  is  also  zero,  and 
<p'(a)  <p  (a) 

vice  versa. 

fix) 
Similarly,  if  the  true  value  of  -j4  he  infinity  when  x  =  a, 

then  ^T7-r  is  really  zero :  we  have,  therefore,  ^rr4=  o,  by  what 
/(a)  y  /  f(a)  J 

has  been  -just  established ;  .*.    ,,  {  =  oo. 

*  w 

Accordingly,  in  all  cases  the  value  of     //  {*   determines 
that  of  ^4-c  for  either  of  the  indeterminate  forms  -  or  S-. 


0(«)  '"       o 


00 


*  On  referring  to  Art.  69,  the  student  will  observe  that  —~r  is  of  the  form 

<p'(x) 

fix) 
—  whenever   -4-r  =  ^-»  so  that  the  process  given  above  -would  not  seem  to  assist 
00  <p(x)      °°'  f  & 

us  towards  determining  the  true  value  of  the  fraction  in  this  case ;  however,  we 
generally  find  a  common  factor,  or  else  some  simple  transformation,  by  which 

o 
we  are  enabled  to  exhibit  our  expression,  after  differentiation,  in  the  form  -. 

For  example    — — ■ is  of  the  form  -22—  when  x  =  -:    here  fix) 

—  00  2 


log(*-^) 


1  f'(%)  00 

=  sec2#,  <b'(x)  =  — — ,  and  the  fraction  —rri  is  still  of  the  form  — -,  but  it  can 

2 

7T 

X 

2  O 

be  transformed  into        .  ,  which  is  of  the  form  - :  the  true  value  of  the  latter 
cos3a;  o 

fraction  can  be  easily  shown  to  be  —  00  when  x  —  -. 

In  some  instances  an  expression  becomes  indeterminate  from  an  infinite  value 

of  x.    The  student  can  easily  see,  on  substituting  -  for  x,  that  our  rules  apply 

equally  to  this  case. 


Indeterminate  Expressions  of  the  Form  {f(x)}^x\      105 

93.    Indeterminate    Expressions     of   the     Form 

{f{x))W\  Let  u  =  {f(x)}^x\  then  log  u  =  (f>(x)  logf(x). 
This  latter  product  is  indeterminate  whenever  one  of  its  fac- 
tors becomes  zero  and  the  other  infinite  for  the  same  value 
of  x. 

(1.)  Let  (J)(x)  =  o,  and  log  {/(x)}  =  ±  00 ;  the  latter  re- 
quires either  f{x)  =  00,  ot/(x)  =  o. 

Hence,  {/(x))^  becomes  indeterminate  when  it  is  of  the 
form  o°,  or  00 °. 

(2.)  Let  <p(x)  =  ±00,  and  log  {fix)}  =  o,  orf(x)  =  1 ;  this 
gives  the  indeterminate  forms 

i°°  and  i'00. 

Hence,  the  indeterminate  forms  of  this  class  are 

o°,  oo°,  and  i±x. 


Examples. 
I .  (sin  x)  tan  *  is  of  the  form  0°,  when  x  =  o. 

it                            i             A         i      /  •      x      log  (sin  x) 
Here  log  u  —  tan  x  log  (sin  x)  =  — — -. 

COD  *V 

The  true  value  of  this  fraction  is  that  of 
cot  x 


=  —  cos  x  sin  x,  or  o  when  x  =  o. 


—  cosec2# 
Hence  the  value  of  (sin  #)tan  x  =  e°  =  1  at  the  same  time. 

2.  (sin  #)tan  x,  when  a;  =  -. 

2 

This  is  of  the  form  1  •,  but  its  true  value  is  easily  found  to  be  unity. 

1 

3.  ( :z:::: ) x  ,  when  x  =  o. 


Here 


,    .  tana;  x2 

but  s  r  +  —  +  &c. 

x  3 


io6  Indeterminate  Forms. 

.      tan  x  (       x*  \      x2 

.*.  log =  log  [  I  +  — 1-  &c.    =  —  +  &c. 

*  \        3  /      3 

Hence,  the  true  value  of  log  u  is  -  when  a;  =o ;  and,  accordingly,  the  value  of  u 

3 
is  e$  at  the  same  time. 

4*  «  =  [  i  +  -  ]  ,  when  x  =  o. 

t  x                                       x     ,                   log  (i  +  «z) 
-Let  #  =  -,  then  log  u  =  — E-i ; 


a 


.'.  by  Art.  92,  the  true  value  of  log  u  when  3  =  ©o  is  that  of ,  or  is  zero. 

1  +  az 
Hence,  the  value  of  u  is  1  at  the  same  time. 


-(-3" 


5.  u  =  [ 1  -\ —  )   ,  when  x  =  00. 


tm-  r    4.x.      i  log(i  +  «z) 

Let  x  =  -,  then  log  w  = 


Z  "  Z 


the  true  value  of  which  is  a  when  z  is  zero. 

Hence,  the  true  value  of  wis  ea\  as  also  follows  immediately  from  Art.  29. 

/  I  \  tan  x 

6.  I  -  I        ,  when  x  =  o.  ^4ws.   1 . 


(j  \  tan  x 
x)         ' 


H) 


tan-r— 

a ,  when  x  =  a. 


94.  Compound  Indeterminate  Forms. — If  an  in- 
determinate form  be  the  product  of  two  or  more  expressions, 
each  of  which  becomes  indeterminate  for  the  same  value  of  a?, 
its  true  value  can  be  determined  by  considering  the  limiting 
value  of  each  of  the  expressions  separately ;  also  when  the 
value  of  any  indeterminate  form  is  known,  that  of  any  power 
of  it  can  be  determined.  These  are  evident  principles  :  at 
the  same  time  the  student  will  find  them  of  importance  in  the 
evaluation  of  indeterminate  functions  of  complex  form.  We 
will  illustrate  their  use  by  a  few  elementary  applications. 

Examples. 
1.  Find  the  value  of 

/  7T  -  2X  \  «  IT 

xm  (sin  x) tan  *    — ]   ,  when  x  =  -. 

\2  sin  2X)  2 

(7r\  m 
-  1    ,  and  that  of  (sin  x) tan*  is  unity :  see  p.  105. 


Examples.  107 

Again, becomes  — : on  substituting z  for  x :  hence  its  true 

2  sin  2x  2  sin  zz  2 

value  is  -  -when  2  =  0. 
2 

IT  ■7TWI 

Accordingly,  the  true  value  of  the  proposed  expression  when  x  =  -  is  —^. 

2.  —  when  #  =  00. 

/  x\  n  X 

This  fraction  can  be  written  in  the  form  /  —  \  .  The  true  value  of  ~,  by  the 


1 

c 

—  en 


method  of  Art.  92,  is  that  of  ■"£•;  but  the  value  of  the  latter  fraction  is  zero 


when  x  =  00  ;  hence  the  true  value  of  the  proposed  fraction  is  also  zero  at  the 
same  time. 

3.  u  =  xn  (log  x)m,  when  x  =  o,  and  m  and  «  are  positive. 


Here  w  =  (#OT  log  x)m% 

log  # 


is  of  the  form  -— -  when  x  =  o ; 


I 

» 

X 

or 

-  mxm 

n 

X 

-^-1 

m 

n 

m 

its  true  value  is  that  of 


Hence,  the  true  value  of  the  given  expression  is  zero. 

This  form  is  immediately  reducible  to  the  preceding,  by  assuming  xn  =  &~v. 

4*  u  = when  x  =  00. 

b*n 


Here  « 


'  \b*nm) 


but  if  b  >  1,  and  n>  in,  bxK~m  =  00  when  x  =  00.     Consequently  the  value  of  wis 
of  the  form  o00 ,  or  is  zero  in  this  case. 

Again,  if  m  >  n,  bx11  m  =  o  when  x  =  oo}  and  the  true  value  of  u  is  00. 


io8  Indeterminate  Forms. 


a  x 
u  = r  "when  x  =  o. 


Let  x  =  -,  and  this  fraction  is  immediately  reducible  to  the  form  discussed  in 

z 

the  previous  Example. 

(i  -  cos#Wlog(i  +x)}m      .  ,        i 

6. '  }    &v — .  when  x  =  o.  -4rcs.  — . 

(r  +  #)*  -e 

7.  w  = ,  when  x  =  o. 

From  Art.  29,  this  is  of  the  form  - ;  to  find  its  true  value,  proceed  by  the 

o 

method  of  Art.  90,  and  it  becomes 

1  (X  -  (1  +  x)  log  (1  +  x)) 

Again,  substituting  for  (1  +  x)x  its  limiting  value  e,  we  get 

(a;  -  (r  +  a;)  log  (1  +  x))  ^ 
e[  x*(i+x)  y 

the  true  value  of  which  is  readily  found  to  be  —  when  x  =  o.  Compare  Ex.  29, 

p.  94. 

\  mx  —  1 


8. 


sin  x 


(a  sm  x  —  smax)n      . 

{ — :  >  ,  when  x  =  o. 

(x(cos#  —  cos«#)) 


The  true  value  of  — : ,  when  x  =  o,  is  log  m ; 

a  sin  a;  —  sin  ax      , 

and  that  of  -. :,  when  x  =  o, 

#(cos#  —  cos##) 

has  been  found  in  Example  6,  Art  89,  to  be-;  hence  the  true  value  of  the  given 

3 


(a\  n 
-  J    log  m. 


8. 


Examples.  109 


Examples. 


[  x  +  sin  7.x  —  6  sin  -  J 
f  4  +  cos  x  -  5  cos  -  J 


a:  =  o. 


x  =  n.  00 . 


<t>(x)-4>(a)  <P  \a) 

/sin  nx\  m 

2-  (-)  • 

cos  #0  —  cos  M0 
3*        (z3  -  w2)»-    ' 

V  «  +  a;  —  a,/2^ 

4-    , 7=* 

V  co  +  3a;  -  2y"  x 

#n+l  _  #w+l 


e*  —  erx  -  2x 
'       ie*- if     ' 


1  —  sin  #  +  cos  x 
sin  #  +  cos  a;  —  1 

tan  x  —  sin  .r 


X  =  «. 

*s/z. 

»  =  -  I. 

los  ©■ 

a;  =  0. 

I 
3* 

a?  =  -. 
2 

I. 

a?  =  0. 

I 
2* 

x  —  a. 

\/  ia 

sin3#       ' 
(g*  -a2)*+(fl-a;)t 

xl  tan  x 

10.  -,  a;  =  o.  1. 

(ex-i)2 

as'mx—a  ir 

11.  . : — ,  x=  -.  a  log  a. 

log  sm  #  2 

w  /#\ 

12.  — cot  I  - ),  #  =  q.  o. 
x            \nj 

x2  +  2  cos  #  -  2  1 

13.  7t ,  *-».  -. 


2 


(!) 


no 


Examples, 


i5- 

1 6. 

17. 
18. 


\A  + 


cos  ix  —  sma; 


x  sin  ^x  +  x  cos  x 

xa  sin  na  —  «a  sin  a;« 
tan  na  —  tan  a;« 


when  x  =  -.    -,4ms.  . 

2  3ir 


a;  =  n. 
na~l  (n  cos  na  -  sin  «#)  cos2wff. 


tan  nx  —  n  tan  x 


1  —  cos  m#    nsmx  —  sin  rc# 
(2  sin  a?  —  sin  2#)2 


(sec  a;  —  cos  2a;)3  ' 
19.   a;1"*, 


x  =  o. 


a;  =  o. 


a;  =  1. 


(a;  -  y){0'(a?)  +  <ft'(y)}  -  24  (x)  +  a»(y) 

20.  :  rr  ,     x  — 

(x  -  y)3 


a; log  (l  +  x) 
1  —  cos  x  ' 


22.   x  .  e«, 

ex  —  e~x 


23 


'  log(i+aj)' 

nx  —  I  ir 

24"      2a;2     +  (e2^  -  1)  x* 

log  (tan  2a?) 
5'    log  (tan  a;)' 

ex  +  log  (1  -  x)  —  1 


26. 


tana;  -  a; 


a;  =  o. 


x  =  o. 


a;  =  o. 


#  =  o. 


a;  =  o. 


a?  =  o. 


_4 

m2' 

8 
125* 

1 
0" 

4>'"(y) 

6     ' 


2. 
00 . 
2. 

2  " 

I. 

I 
2* 


27 


2m 

■  (i-^)vr^m2 


tan 


.1  \/ 1  -  m2 


m 


1  +  m 

cos  d>  -    cos  d>,    m  =  1. 

T       1  -  w 

cos  3^) 


28. 


log  (1  +  X  +  x2)  +  log  (I  -  X  +  X2) 


sec  x  —  cos  x 


/aix  +  a2*  +  .  .  .  fl^X  ~ 
■9-  \ n )  ' 


x  =  o. 


a?  =  o. 


a\  a-i  .  , •  «w 


Examples.  1 1 1 

/log  x\  x 

30.  {  — — j,  when  a;  =00.     Ans.  1. 

.  i  ex 

(1  +  x)x-e  +  — 

31.  - ,  #  =  0. 


33-  *2  (1  +  ^j-^3log  (1  +  £j, 


log(i  +  #)    ' 

"  —  e~x  —  ix 
'*      tan  #  —  a;   ' 


«?    tax2  +  bx  +  c\ 
dx  \    a\x  +  b\    )  ' 


#  =00. 


a;  =  00. 


24 


sin  #  —  log  [ex  cos  a?)  1 

32. Li 2,  a;  =  o. 


2 


8* 


1  —  a;  +  log  a; 

34-   7  ->  * =  1.  -  1. 

I  -  \/  2X  —  xz 
X2  —  X 

35-    ; »  x=i.  00. 

I  -  #  +  log  X 

xx  -x 

36.    — ,  *=I.  -2. 

I  —  #  +  log  X 

cos  #  —  log  (1  +  x)  +  sin  x  —  1 

37-  ,       ~s »  x  =  o.  o. 

ex-(i+x) 

ex  +  sin  x  —  1 
38»  1 — 1 — ; — 7-9  x  =  o  2. 


ex  _  e-x  _  2a; 

39-  -z— - — --  #  =  o.  1. 


a 


a  —  x  —  a  log  I  -  J 

41.  ■  ?  3.  _  a.  _  lm 

a  —  y  2ax  —  x* 

tan  (#  +  #)  —  tan  («  —  x)  1  +  a2 

42. - x  =  o  

tan-1  (a  +  x)  —  tan-1  («  —  x)  *  cos3#  ' 

xz  —  3a;  +  2 
43-  £4_6*»  +  8*-3'  *=I- 


1 1 2  Examples. 

44.  (sin a;)8"1*,  when  x  =  o.        Am.  1. 

45.  (sec#)Mseca!,  #  =  o.  1. 

tan*  Tj. 

46.  (sin  x)       ,  #  =  -•  1. 

2 

47.  Find  the  value  of 

(x-y)an+  (y  -a)xn  +  (a-  x)yn  n.n-  1  an-2# 

(#  -  y)  {y  -a)  {a-  x)  *  1.2 

when  x  —  y  —  a. 

Substitute  a  +  h  f or  #,  and  0  +  &  for  y,  and  after  some  easy  transformations  we 
get  the  answer,  on  making  h  =  o,  and  k  =  o. 

#  +  tan  #  —  tan  2x  .        7 

4.3.  ,  x  =  o.       Ans.  -p< 

T      2#  +  tan  a;  -  tan  3#  20 

a;  +  sin  x  —  sin  2#  —  7 


49 


#  =  0. 


20;  +  tan  x  —  tan  30;'  52 


\/x  —  \/a  +  '\/x  —  a 

50. .  ,  ,— 

^/a;2  _  a2  ^/  20 


%  =  a. 


2  •  r* 

# sin  a;  —  tan  ar 

3  1 


—  1 


ci  1  2-0.  — 

51,  #5  »  20 


(      »3     ) 


CHAPTEE  Y. 

PARTIAL   DIFFERENTIAL    COEFFICIENTS    AND   DIFFERENTIATION 
OF  FUNCTIONS  OF  TWO  OR  MORE  VARIABLES. 

95.  Partial  Differentiation. — In  the  preceding  chap- 
ters we  have  regarded  the  functions  under  consideration  as 
depending  on  one  variable  solely ;  thus,  such  expressions  as 

eax,  sin  bx,  xm,  &c, 

have  been  treated  as  functions  of  x  only ;  the  quantities 
a,  b,  m,  .  . ;...  being  regarded  as  constants.  We  may,  however, 
conceive  these  quantities  as  also  capable  of  change,  and  as 
receiving  small  increments  ;  then,  if  we  regard  x  as  constant, 
we  can,  by  the  methods  already  established,  find  the  differen- 
tial coefficients  of  these  expressions  with  regard  to  the  quan- 
tities, a,  b,  m,  &c,  considered  as  variable. 

In  this  point  of  view,  eax  is  regarded  as  a  function  of  a  as 
well  as  of  x,  and  its  differential  coefficient  with  regard  to  a 

d  (eax) 
is  represented  by  ,  or  x  eax  by  Art.  30 ;  in  the  derivation 

of  which  x  is  regarded  as  a  constant. 

In  like  manner,  sin  {ax  +  by)  may  be  considered  as  a 
function  of  the  four  quantities,  x,  y,  a,  b,  and  we  can  find  its 
differential  coefficient  with  respect  to  any  one  of  them,  the 
others  being  regarded  as  constants.  Let  these  derived  functions 
be  denoted  by 

du     du     du     du 

dx'    dy'     da9     db* 

respectively,  where  u  stands  for  the  expression  under  con- 
sideration, and  we  have 

du  du  .  . 

—  =  a  cos  {ax  +  by),     —  =  b  cos  (ax  +  by), 

du  ,         _  x       du  .         .  . 

—  =  x  cos  (ax  +  by),     —  =  y  cos  (ax  +  by). 


1 1 4  Partial  Differentiation. 

These  expressions   are   called  the  partial  differential  coef- 
ficients of  u  with  respect  to  x,  y,  a,  b,  respectively. 
More  generally,  if 

/0,  y,  *) 

denotes  a  function  of  three  variables,  x,  y,  z,  its  differential 
coefficient,  when  x  alone  is  supposed  to  change,  is  called  the 
partial  differential  coefficient  of  the  function  with  respect  to  x; 
and  similarly  for  the  other  variables  y  and  %.  If  the  function 
be  represented  by  u,  its  partial  differential  coefficients  are 
denoted  by 

du      du      du 

dx9     dy*     dz ' 

and  from  the  preceding  it  follows  that  the  partial  derived 
functions  of  any  expression  are  formed  by  the  same  rules  as 
the  derived  functions  in  the  case  of  a  single  variable. 

Examples. 

1.  u  =  {ax*  +  by*  +  ezz)«. 

Here  —  =  2 nax  (ax2  +  by2  +  cr£  V1"1, 

dx 

—  =  2nby  (ax2  +  by2  +  cz2)n~l, 
dy 

—  =  2ncz  (ax2  +  by2  +  cz2)n'K 


y 

du             1              du            —  x 

a/V2  ~x<1     y    y*y y2—%2 

3- 

du            ,         du 
u  =  xv,         —  =  yxv-1,       —  =  xv 
dx                      dy 

4- 

u  =  x-<p  (xy) . 

du              .     .        „     ,  .     . 
—  =  2Xcp  (xy)  +  x2y<p  (xy). 
ax 

Differentiation  of  a  Function  of  Two  Variables.         115 

96.  Differentiation  of  a  Function  of  Two  Vari- 
ables.— Let  u  =  $  (x,  y),  and  suppose  x  and  y  to  receive  the 
increments  h,  k,  respectively,  and  let  Au  denote  the  corre- 
sponding increment  of  u,  then 

Au  =  (j)  (x  +  h,  y  +  k)  -  0  (x,  y) 

=  $  (x  +  h,  y  +  k)  -  <p  (x,  y  +  k)  +  0  (x,  y  +  k)  -  $  (x,  y) 

=  $  (x  +  h,  y  +  k)  -  <j>  {x,  y  +  k)        ft  (x,y  +  k)  -ft  (x,  y) 
h  k 

If  now  h  and  k  be  supposed  to  become  infinitely  small, 
by  Art.  6  we  have 

ft  (x  +  h,  y  +  h)  -  ft  (x,  y  +  k)      d  .  ft  (x,  y  +  k) 


and 


h  dx 

ft  (x,  y  +  k)  -  ftp,  y)     d  .  ft  (x,  y) 
k  dy 


In  the  limit,  when  k  is  infinitely  small,  $  (x,  y  +  k) 
becomes  ft  (x,  y),  and 

rf-»(W*)  becomes  <L*p*l} 

dx  dx 

hence  we  get,  neglecting  infinitely  small  quantities  of  the  second 

order, 

du  7     du  _ 
du  =  —  h  +—  k, 
dx        dy 

where  h  and  k  are  infinitely  small. 

If  dx,  dy,  be  substituted  for  h  and  k,  the  preceding 
becomes 

du  du  7  ,  . 

du  =  —  dx  +  —  dy.  ( 1 ) 

dx  dy    *  ' 

In  this  equation  du  is  called  the  total  differential  of  w, 
where  both  x  and  y  are  supposed  to  vary. 

The  student  should  carefully  observe  the  different  mean- 
ings given  to  the  infinitely  small  quantity  du  in  this  equation. 

Thus,  in  the  expression  —  dx,  du  stands  for  the  infinitely 

1  2 


1 1 6  Partial  Differentiation. 

small  change  in  u  arising  from  the  increment  dx  in  x,  y  being 

regarded  as  constant.     Similarly,  in  —  dy,  du  stands  for  the 

infinitely  small  change  arising  from  the  increment  dy  my,  x 
being  regarded  as  constant.  If  these  partial  increments  be 
represented  by  dxu,  dyu,  the  preceding  result  may  be  written 
in  the  form 

du  =  dxu  +  dyu. 

That  is,  the  total  increment  in  a  function  of  two  variables  is 
found  by  adding  its  partial  increments,  arising  from  the 
differentials  of  each  of  the  variables  taken  separately. 

Examples. 

i.  Let  x  =  rco&d,  in  which  r  and  6  are  considered  variables,  to  find  the 
total  differential  of  x. 


Here 

dx                   dx 

—  =  cos  6,    — -  =  -  r  sm  6. 

dr              '    dd 

Hence 

dx  =  cos  6  dr  -  r  sin  6  dd. 

2. 

x%      y% 

U  =  ~2+  T' 

Here 

du       ix      du       zy 
dx        a2'     dy       b%  ' 

2x  _       iy  _ 
.  •.  du  =  —dx  +  -—  dy. 

-  1 .     Let  -  =  z,  then  u  =  <t>  (z), 

y)  y 

*0- 


du       du  dz 
dx       dz  dx 

du      du  dz 

dy       dz  dy  y1 

ydx  —  xdy 


-o 


■••*-*€) 


t 


Again,  multiplying  the  former  of  the  two  preceding  equations  by  x,  and  the 
latter  by  y,  and  adding,  we  get 

du        du 
x—  +  y—  =  o. 
dx        dy 


Differentiation  of  a  Function  of  Three  or  more  Variables.    117 

97.  Differentiation  of  a  Function  of  Three  or 
more  Variables. — Suppose 

u  =  $  (x,  y,  a), 

and  let  h,  k,  I  represent  infinitely  small  increments  in  x,  y,  z, 
respectively;  then 

Au  =  (j>  (x  +  h,  y  +  Jc,  z  +  I)  -  (j)  (xy  y,  z) 

(j>  (x  +  h,  y  +  h,  z  +  T)  -  0  (x,  y  +  k,  z  +  I) 

(j)  (x,y  +  k,z+l)  -  0  (x,y,z  +  l)       <j>  {xiy,z+l)-<l>(x,yiz) 
+  -k  k+  f  I, 

which  becomes  in  the  limit,  by  the  same  argument  as  before, 
when  dx,  dy,  dz,  are  substituted  for  h,  k,  I, 

du  du  _       du  T  ,  . 

du  =  —  dx  +  —  dy  +  —  dz.  (2) 

dx  dy  dz 

Or,  the  infinitely  small  increment  in  u  is  the  sum  of  its 
infinitely  small  increments  arising  from  the  variation  of  each 
variable  considered  separately. 

A  similar  process  of  reasoning  can  be  easily  extended  to 
a  function  of  any  number  of  variables  ;  hence,  in  general,  if 
u  be  a  function  of  n  variables,  xx,  x%,  xz,  .  .  .  xn, 

7       du    7        du  du   -  .  N 

du  =  —  dxi  +  —  dx2  +  . . .  +  —  dxn.  (3) 

ax\  aXi  aXifi 

98.  If 

u  =  f(v,  w), 

where  v,  w,  are  both  functions  of  x  ;  then,  from  Art.  96,  it  is 
easily  seen  that 

du     df(v,  w)  dv     df(v,  w)  dw 
dx  dv      dx  dw      dx' 

This  result  is  usually  written  in  the  form 

du     du  dv      du  dw  . 

dx      dv  dx     dw  dx'  *  ' 

In  general,  if 


1 1 8  Partial  Differentiation. 

where  ylf  y2,  .  .  .  yn,  are  each  functions  of  x,  we  have 

du      du  dyi      du  dy%  da  dyn 

dx      dyl  dx      dy%  dx  dyn  dx' 

Also,  if  yl9  y2,  &c,  yn9  be  at  the  same  time  functions  of 
another  variable  s,  we  have 


and  so  on. 


du  _  du  dyx      du  dy2  du  dyn 

dz      dyx  dz      dy2  dz  dyn  dz 


Examples. 


1.  Let  u  =  <j)(Xt  T)t 

where  X  =  ax  +  by,         Y  =  a'x  +  b'y ; 

,,                                  du        du  dX      du  d¥ 
then  —  = f. 

dx       dX  dx      dY  dx 

du        du  dX      du  dY 
dy       dX  dy      dY  dy  ' 

.  dX  dX       ,      dY  dY       „ 

hut  _  =  a ,     ■—  =  b,     —  =  a  ,     —  =  b  . 

dx  dy  dx  dy 

-rr  du  du        .du 

Hence  —  =  a h  a  -=. , 

dx         dX        dY' 

du  du         du 

dy         dX        dY' 

2.  More  generally,  let 

u  =  <j>(X,Y,Z), 

where  X  =  ax  +  by  +  cz, 

Y  =  a'x  +  b'y  +  c'z, 

Z  =  a"x  +  b"y  +  c"z. 

"When  these  substitutions  are  made,  u  becomes  a  function  of  x,  y,  z,  and  we 
have 

du         du        ,  du        „  du 

Tx  =  adx  +  adY+a  dz> 

du       y  du  du  du- 

dy-=bdX+bdY+b   12 

du  du        .du        „  du 

dz         dX^     dY         dZ' 


Differentiation  of  a  Function  of  Differences. 


1 19 


98*.  ^Differentiation  of  a  Function  of  diffe- 
rences.— If  u  be  a  function  of  the  differences  of  the  vari- 
ables, a,  j3,  y  :  to  prove  that 

du     du      du 

\-  —^-\ =  0. 

da      dp      dy 

Let  a  -  ft  =  x,  j3  -  y  =  y,  y  -  a  =  z;  then,  u  is  a  function 
of  x}  y}  %  \  and,  accordingly,  we  may  write 


u  =  <j>  (x,  y,  z). 

du      du  dx     du  du      du  dz      du      du 
da      dx  da      dy  da      dz  da      dx      dz 

du      du      du  du      du     du 

djd      dy      dx'         dy      dz      dy ' 

du      du      du 
'  da      dj5      dy 

This  result  admits  of  obvious  extension  to  a  function  of 
the  differences  of  any  number  of  variables. 


Hence 
Similarly, 


1.  If 


2.  If 


dA      dA      dA 
da       dp       dy 


* 

I, 

h 

I, 

I, 

A  = 

a, 
a~, 

ft 

7, 
72, 

5, 
53, 

a3, 

03, 

73, 

53, 

dA      dA 
—  +  — 

da       dfi 

dA 
dy 

dA 
+  ~d~8 

=  o. 

I, 

i, 

I, 

I, 

A  = 

a, 
«2, 

ft 

7> 
72, 

5, 

s2, 

a4, 

&, 

74, 

54, 

h 

I, 

I, 

1, 

d8        v 

a, 
a-, 

ft 

02, 

7> 

72, 

8, 

s2, 

a3, 

/33, 

73, 

s3, 

,  prove  that 


,  prove  that 


1 2  o  Partial  Differentiation. 

99.  Definition  of  an  Implicit  Function. — Suppose 
that  y,  instead  of  being  given  explicitly  as  a  function  of  x}  is 
determined  by  an  equation  of  the  form 

/(*>  y)  =  °> 

then  y  is  said  to  be  an  implicit  function  of  x ;  for  its  value,  or 
values,  are  given  implicitly  when  that  of  x  is  known. 

100.  Differentiation   of  an  Implicit  Function. — 

Let  h  denote  the  increment  of  y  corresponding  to  the  incre- 
ment h  in  x,  and  denote  f(x,  y)  by  u. 

Then,  since  the  equation  fix,  y)  =  o  is  supposed  to  hold 
for  all  values  of  x  and  the  corresponding  values  of  y,  we 
must  have 

fix  +  h,y  +  k)  =  o. 

Hence  du  =  o ;  and  accordingly,  by  Art.  96,  we  have, 
when  h  and  h  are  infinitely  small, 

du  7     du  7 

— -  h  +  — -  k  =  o  ; 
ax        ay 

du 

.  k     dy        dx  . 

hence  in  the  limit  -  =  _=_^.  (6) 

dy 

This  result  enables  us  to  determine  the  differential 
coefficient  of  y  with  respect  to  x  whenever  the  form  of  the 
equation  f(x,  y)  =  o  is  given. 

In  the  case  of  implicit  functions  we  may  regard  x  as 
being  a  function  of  y,  or  y  a  function  of  x,  whichever  we 
please — {n  the  former  case  y  is  treated  as  the  independent 
variable,  and,  in  the  latter,  x  :  when  y  is  taken  as  the  inde- 
pendent variable,  we  have 

du 
dx        dy  _    1 
dy        du      dy 

QjJU         (JjJU 

This  is  the  extension  of  the  result  given  in  Art.  20,  and 
might  have  been  established  in  a  similar  manner. 


Differentiation  of  an  Implicit  Function.  121 

Examples. 
1 .  If  a3  +  y3  —  2>axy  —  e,  to  find  — . 

Here  *!.  «  3 (* - <tf,      |  =  3(^-^); 


/S^  Art.  38. 


dy       a;2  —  «2/ 
dx       ax  —  y2' 


2.  It  £+£  =  I,  to  find  ^. 

dw       wiK,n"1      du       my™-1  _     _    dy  _       /%\m~l/b 
dx  ~      am   '     d«/  ~      J»    '    '  "  da;  ~       \y/        \fl 

dy       w  /a;  log  y  —  y\ 

3.  *logy-ylog««o.       ^--^-1__j. 


1 01.  If  w  =  0  (a?,  2/),  where  x  and  ?/  are  connected  by  the 
equation  f(x,  y)  -  o,  to  find  the  total  differential  of  u  with 
respect  to  x ;  y  being  regarded  as  a  function  of  x. 

Here,  by  Art.  98,  we  get 

du     dcf)     d(j)  dy 
dx     dx      dy  dx' 


Also 


df     df  dy 

dx     dy  dx 


Hence,  eliminating  -~,  we  get 
dx 


d(J)  df     df  dcj) 
du      dx  dy      dx  dy  /~\ 

dx  df 

dy 


122 


Partial  Differentiation. 


This  result  can  also  be  written  in  the  following  deter- 
minant form : 

d(j>      d(j> 

dx'     dy 


du 
dx 


df_      df_ 
dx*     dy 


df_ 
dy 


More  generally,  let  u  =  <p  (x,  y,  z),  where  x,  yy  z,  are  con- 
nected by  two  equations, 

/iO,  V,  *)  =  o,      f2(x,  y,  z)  =  o ; 
then,  as  in  the  preceding  case,  we  have 


and  also 


Hence,  we  get 


du  d(p      d(p  dy     d<p  dz 

dx  dx      dy  dx      dz  dx* 

dfx  dfx  dy     dfx  dz 

dx  dy  dx     dz  dx 

df2  df2  dy     df2  dz 

dx  dy  dx      dz  dx 


dcf) 

d(j> 

dcj) 

dx' 

dy' 

dz 

dfx 

df 

dfi 

dx' 

dy' 

dz 

df, 

dx' 

dfz 
dy' 

dh 
dz 

da 
dx 


dy'        az 
This  result  easily  admits  of  generalization. 


(8) 


Miller's  Theorem  of  Homogeneous  Functions.  123 

102.  Eider's    Theorem    of  Homogeneous    Func- 
tions.— If 

u  =  AaP  yq  +  Bx*'  yqf  +  Cap"  yq"  +  &c, 
where 

p  +  q  =  p  +  q  =  p"  +  q"  =  &c.  =  n, 

to  prove  that 

du        du  ,  > 

x  —  +  y-—  =  nu.  (9) 

dx     J  dy  K  J 

(I'll 
Here         x  —  =  Apxp  yq  +  Bp'  xpf  yqf  +  &c.  ; 

ft ?/ 

y  —  =  AqxP  yq  +  Bq  x?'  yq'  +  &c. ; 

.'.  x—  +  y  —  =  A(p  +  q)xPyq  +  B(pr  +  q')  xp'  yq'  +  &c. 
dx        cty 

=  nAxp  yq  +  nBxpf  yqf  +  &o.  =  nu. 

Hence,  if  u  be  any  homogeneous  expression  of  the  nth 

degree  in  x  and  y,  not  involving  fractions,  we  have 

• 

du        du 
x  -=-+  y  —  =  nu. 
dx        dy 

Again,  suppose  u  to  he  a  homogeneous  function  of  a 
fractional  form,  represented  by  —  ;  where  0i,  02,  are  homo- 

02 

geneous  expressions  of  the  nth  and  mth  degrees,  respectively, 
in  x  and  y ;  then,  from  the  equation 


we  have 


and 


du 

0i 
u  =  — 

02 

d(j}2 
dx 

dx 
du 

(03)2 
C?0i 

^Hy~* 

d(f>2 

dy 

(<t>*f 

124  Partial  Differentiation, 

accordingly  we  get 

/  dtyi        d£A  _      (    dfa        dfc 

du        du  \  dx         dy  )         \    dx         dy 

XTx  +  ydy=  ^p  ' 

but,  by  the  preceding, 

dd)i         d(bi  dd)2         dd)2 

.                              du        du     n<bx  62  -  md>i  d>2 
h6nCe  'H  +  '*" (^ 

=  in  -m)  —  =  (n  -  m)  u\ 

which  proves  the  theorem  for  homogeneous  expressions  of  a 
fractional  form. 

This  result  admits  of  being  established  in  a  more  general 
manner,  as  follows : 

It  is  easily  seen  that  a  homogeneous  expression  of  the  nth 
degree  in  x  and  y,  since  the  sum  of  the  indices  of  x  and  of  y 
in  each  term  is  n,  is  capable  of  being  represented  in  the 
general  form  of 

Accordingly,  let  u  =  xn  $  I  -  J  =  xnv, 

where  v  =  <p  I  - 

mi  du         m.         ndv 

Then  —  =  nxn~^v  +  xn  — , 

\AiJU  iXJL- 

du         dv 
and  —  =  xn  — : 

dy         dy 

multiply  the  former  equation  by  x,  and  the  latter  by  y,  and 
add ;  then 

du        du  ..  f  dv        dvs 


x  —  +  y  —  =  nxnv  +  xn  I  x  —  +  y  —  ) ; 
dx     *  dy  \  dx     J  dyj ' 


Enter's  Theorem  of  Homogeneous  Functions.  125 


but  (by  Ex.  3,  Art.  96), 

dv        dv 
dx        dy 

du        du 

hence          x  —  +  y  —  =  nxnv  = 

dx        dy 

=  nu, 

which  proves  the  theorem  in  general. 

In  the  case  of  three  variables,  x,  y,  2, 

suppose  u  =  Ax?  yq  zr, 

then  we  have 

du       .  du       .  du       . 

x—~=  Ap xP yqzr,     y—  -  Aq xp yq zr,      z  —  =  Ar xp yq z7  ; 
dx  cly  ciz 

du        du        du       . .  .    m   M  m      ,  N 

.'.x  —  +  y—  +  z—  =  A(p  +  q  +  r)  xp  yq  zr  =  ( p  +  q  +  r)  u  ; 
ctx        ay        ciz 

and  the  same  method  of  proof  can  be  extended  to  any  homo- 
geneous function  of  three  or  more  variables. 

Hence,  if  u  be  a  homogeneous  function  of  the  nth  degree 
in  x}  y,  z,  we  have 

du        du        du  .     N 

x  —  +  y  —  +  z—  =  nu.  ( 1  o) 

dx        dy        dz 

It  may  be  observed  that  the  preceding  result  holds  also 
if  n  be  a,  fractional  or  negative  number,  as  can  be  easily  seen. 

This  result  can  also  be  proved  in  general,  by  the  same 
method  as  in  the  case  of  two  variables,  from  the  considera- 
tion that  a  homogeneous  function  of  the  nth  degree  in  x,  y,  z 
admits  of  being  written  in  the  general  form 


u  =  xn  6  [  -,  -  I, 

•     »   /y      /y  1 


or  in  the  form 


y  z 

u  =  xn  0  (v,  w),  where  v  =  -,  and  w  =  -. 

X  X 

Proceeding,  as  in  the  former  case,  the  student  can  show, 


126  Partial  Differentiation. 

without  difficulty,  that  we  shall  have 

du        du        du 
x~  +  y  —  +  g  —  =  nu. 
ax        dy        dz 

Another  proof  will  be  found  in  a  subsequent  chapter,  along 
with  the  extension  of  the  theorem  to  differentiations  of  a 
higher  order. 

Examples. 

Verify  Euler's  Theorem  in  the  following  cases  by  direct  differentiation : — 

x3  +  y3  du         du        Ku 

i.  u  =  ; rr ;         prove  x  —  +  y  —  =  —• 

(x  +  y)i'  F  dx      J dy         2 

x3  +  ax2y  +  by3  du        du 

U=      a>x>  +  by     '    »     Xdx  +  ^d~y  =  U' 

.    .  x2  —  y2  du        du 

/x3  —  y3\  du        du 

103.  Theorem. — If  U  =  u0  +  ux  +  u%  .  .  .  +  un, 
where  u0  is  a  constant,  and  ux,  u2,  .  .  .  un,  are  homogeneous 
functions  of  x,  y,  z,  &c,  of  the  ist,  2nd,  .  .  .  nth  degrees, 
respectively,  then 

dU       dU       dU 
2~a^  +  1/'ay+Z'dz+''':=Ul  +  2U2  +  3U3+'''  +  nUn'      (*  ^ 

For,  by  Euler's  Theorem,  we  have 

aUy        a%vf        aWiy 

■ \-  y +  % 

dx         dy         dz 

since  ur  is  homogeneous  of  the  rth  degree  in  the  variables. 

Con.  If  TT  =  o,  then 

dU       dU       dU  .     x 

X~dx:  +  yd^  +  Zlz''":=~  ^Un-1  +  2%~2  +  '  *  *  +  m^'      ^ 

This  follows  on  subtracting 

nii0  +  nux  +  .  .  .  +  nun  =  o 

from  the  preceding  result. 


tv  Ivy*  Lb  Ivy  \Af  tl'y  q 

x  —  +  y  —-  +  z-=-  +  &c.  =  run 


Remarks  on  JEuler's  Theorem.  1 2  7 

104.  Remarks  on  JEuler's  Theorem. — In  the  appli- 
cation of  Euler's  Theorem  the  student  should  be  careful  to 
see  that  the  functions  to  which  it  is  applied  are  really 
homogeneous  expressions.      For  instance,  at  first  sight  the 

expression  sin-1  ( — — '—  J  might  appear  to  be  a  homogeneous 

function  in  x  and  y ;  but  if  the  function  be  expanded,  it  is 
easily  seen  that  the  terms  thus  obtained  are  of  different 
degrees,  and,  consequently,  Euler's  Theorem  cannot  be 
directly  applied  to  it.     However,  if  the  equation  be  written 

/y>   -J-    if/ 

in  the  form  -= r  =  sin  u,  we  have,  by  Euler's  formula, 

xi  +  yh  '  '     J  9 

d  sin  u        cl  sin  u     sin  u 

x  — = h  y  — = —  = , 

dx  dy  2 

du        du\      sin  u 


or  cos  u  \x  —  +  y  —    =  —  ; 

\    dx        dy  J         2 

.  du        du      tan  u      \  x  +  y 

hence  x  —  +  y  —  = 


dx        dy         2  2  </(«*  + y*)*  -  (a>  +  y)«" 

When,  however,  the  degrees  in  the  numerator  and  the 
denominator  are  the  same,  the  function  is  of  the  degree  zero, 
and  in  all  such  cases  we  have 

du        du 
x—+y—  =  o.     ■ 
dx        dy 


1     ,       .  ,JL\  hx 


T2  ,        .         xs  +  ya\  x  +  y   —     . 

For  example,  sm"1    -x — —.  ,   tan"1 -,  &,  &c,  may  be 

treated  as  homogeneous  expressions,  whose  degree  of  homo- 
geneity is  zero.     The  same  remark  applies  to  all  expressions 

which  are  reducible  to  the  form  $  (  -  ) ;   as  already  shown  in 
Ex.  3,  Art.  96. 

105.  If  x  =  r  cos  0,  y  =  r  sin  0, 

to  prove  that  xdy  -  ydx  =  r2d0.  (13) 


1 2  8  Partial  Differentiation, 

In  Ex.  i,  Art.  96,  we  found 

dx  =  cos  Odr  -  r  sin  QdQ  ; 
similarly  dy  =  sin  Odr  +■  r  cos  06?0. 

Hence  a?%  =  r  cos  9  sin  0^r  +  r2  cos2  OdO, 

ydx  =  r  cos  0  sin  0<#r  -  r2  sin2  0^0 ; 
.*.  xdy  -  ydx  =  r2  dO. 

106.  If  a?  and  y  have  the  same  values  as  in  the  last,  to 
prove  that 

(dx)2  +  (dy)2  =  (drf  +  r2  (dO)2.  (14) 

Square  and  add  the  expressions  for  dx,  dy,  found  above, 
and  the  required  result  follows  immediately. 

The  two  preceding  formulae  are  of  importance  in  the 
theory  of  plane  curves,  and  admit  of  being  easily  established 
from  geometrical  considerations. 

107.  If        u  =  ax2  +  by2  +  cz2  +  zfyz  +  2gzx  +  ihxy, 

to  find  the  condition  among  the  constants  that  the  same  values  of 
x,  y,  z  should  satisfy  the  three  equations 


Here 


du 

—  =  2gx  +  2jy  +  icz  =  o. 

Hence,  eliminating  x,  y,  %  between  these  three  equations, 
the  required  condition  is 

abc  -of2-  bg2  -  ch2  +  2fgh  =  o ; 

or,  in  the  determinant  form, 

a       h      g 

h       b      f 

9      f      c 


du 
dx 

0, 

du 
dy 

0, 

du 
dz 

0. 

du 
dx 

2  ax 

+  2hy 

+  2gz 

=  0, 

du 

dy 

2hx 

-I-  2by 

+  2jk* 

=  0, 

Remarks  on  Ruler's  Theorem.  129 

The  preceding  determinant  is  called  the  discriminant  of  the 
quadratic  expression,  and  is  an  invariant  of  the  function ;  it 
also  expresses  the  condition  that  the  conic  represented  by 
the  equation  u  =  o  should  break  up  into  two  right  lines. 
(Salmon's  Conic  Sections,  Art.  76.) 

The  foregoing  result  can  be  verified  easily  from  the  latter 
point  of  view ;  for,  suppose  the  quadratic  expression,  u,  to  be 
the  product  of  two  linear  factors,  X  and  Y ; 

or  u  =  XY, 

where  X  =  Ix  +  my  +  nz,      Y  =  Vx  +  m'y  +  n'z  ; 

then  £  =  X~  +  T^  =  I'X  +  IT, 

ax  ax  ax 

du     ^.dY     rrdX       ,^.        ^ 

—  =  A  —  +  F  —  =  mX  +  mY, 
dy  dy  dy 

au      ._  a  jl       __  u>JL       ,  __        ._ 

—  =  A  —  +  F  —  =  nX  +  nY. 
dz  dz  dz 

Here  the  expressions  at  the  right-hand  side  become  zero  for 
the  values  of  x,  y,  z,  which  satisfy  the  equations  X  =  o,  F=  o, 

or  Ix  +  my  +  nz  =  o,         Vx  +  m'y  +  n'z  =  o. 

Hence  in  this  case  the  equations 

du  du  du 

dx       '     dy       '     dz 

are  also  satisfied  simultaneously  by  the  same  values. 

We  shall  next  proceed  to  illustrate  the  principles  of 
partial  differentiation  by  applying  them  to  a  few  elementary 
questions  in  plane  and  spherical  triangles.  In  such  cases  we 
may  regard  any  three*  of  the  parts,  a,  b,  c,  A,  JB,  C,  as  being 


*  The  case  of  the  three  angles  of  a  plane  triangle  is  excepted,  as  they  are 
equivalent  to  only  two  independent  data. 

K 


130  Partial  Differentiation. 

independent  variables,  and  each  of  the  others  as  a  function  of 
the  three  so  chosen. 

108.  Equation  connecting  the  Variations  of  the 
three  Sides  and  one  Angle. — If  two  sides,  a,  b,  and  the 
contained  angle,  C,  in  a  plane  triangle,  receive  indefinitely 
small  increments,  to  find  the  corresponding  increment  in  the 
third  side  c,  we  have 

&  =  a2  +  ¥  -  lab  cos  C; 

.*.  cdc  =  (a  -  b  cos  C)  da  +  (b  -  a  cos  C)  db  +  ab  sin  CdC; 

but      a  =  b  cos  C  +  c  cos  B,      b  =  a  cos  0  +  c  cos  A. 

Hence,  dividing  by  c,  and  substituting  c  sin  B  for  b  sin  C, 
we  get 

do  =  cos  B  da  +  cos  A  db  +  a  sin  B  dC.  (15) 

Otherwise  thus,  geometrically. 

By  equation  (2),  Art.  97,  we  have 

dc    7        tfc    77       die    7~ 

dc  =  —  da  +  —  do  +  — ^  du. 
da  do  Oj\j 

dc 
Now,  in  the  determination  of  —  we  must  regard  b  and  C  as 

aa 

constants ;   accordingly,  let  us  sup- 
pose the  side  CB,  or  a,  to  receive  a 
small  increment,  BB'  or  Aa,  as  in 
the  figure.   Join  AB",  and  draw  B'D 
perpendicular  to  AB,  produced  if 
necessary;    then,  by  Art.  37,  ABr    »  ] 
=  AB  when  BB'  is  infinitely  small,      , 
neglecting  infinitely  small   quanti- 
ties of  the  second  order. 
Hence 

Ac  =  AB'  -AB  =  AD-AB  =  BB; 

dc      ..    .,     ,  Ac      BD  „ 

.-.  —  =  limit  of  —  =  r—-,  =  cos  B. 
da  Aa     BB 


Fig.  4. 


Examples  in  Plane  Trigonometry.  131 

HP 

Similarly,  —  =  cos  A ;  which  results  agree  with  those  arrived 

at  before  by  differentiation. 

dc 
Again,  to  find  -j~.     Suppose  the  angle  C  to  receive  a 

small  increment  AC,  represented  by  £ 

BCB'  in  the   accompanying  figure; 
take  CB'  =  CB,  join  AB',  and  draw 
BB  perpendicular  to  AB ' . 
Then 

Ac  =  AB'  -AB  =  B'B  (in  the  limit) 

=  BB' cos  AB'B  =  BF sin^(7(q.p.).  Fig.  5. 

Also,  in  the  limit,  BB'  =  #(7  sin  BOB"  =  a  AC. 

dc  Ac 

Hence  -t~  =  limiting  value  of  — ^  =  #  sin  B ; 

the  same  result  as  that  arrived  at  by  differentiation. 

In  the  investigation  in  Fig.  5  it  has  been  assumed  that 
AB  -  AB  is  infinitely  small  in  comparison  with  BB;  or  that 

AB-AD 

the  fraction  =p=r —  vanishes  in  the  limit.     For  the  proof 

of  this  the  student  is  referred  to  Art.  37. 

When  the  base  of  a  plane  triangle  is  calculated  from  the 

observed  lengths  of  its  sides  and  the  magnitude  of  its  vertical 

angle,  the  result  in  (15)  shows  how  the  error  in  the  computed 

value  of  the  base  can  be  approximately  found  in  terms  of  the 

small  errors  in  observation  of  the  sides  and  of  the  contained 

angle. 

dC 
ioo.  To    find  -—    when   a   and   0   are    considered 

y  dA 

Constant. — In  the  preceding  figure,  BAB'  represents  the 
change  in  the  angle  A  arising  from  the  change  AC  in  C; 
moreover,  as  the  angle  A  is  diminished  in  this  case,  we  must 
denote  BAB'  by  -  AA,  and  we  have 

BB' -       AB/^A         ABAA       cAA 
sm  AB'B~      cos  B         cosB' 
k  2 


132  Partial  Differentiation, 

Also,  BB'  =  a&C; 

dO      AC  ,.     ,     ..    ...  c 

,.  ^  =  ^(m  the  limit)  —  j^.  (16) 

This  result  admits  of  another  easy  proof  by  differentiation. 

For  #  sin  B  =  b  sin  A  ; 

hence,  when  #  and  b  are  constants,  we  have 

a  cos  B  dB  =  b  cos  -4  *L4  ; 

also,  since  A  +  B  +  C  =  it,  we  have 

dA  +  dB  +  dC  =  o. 

Substitute  for  <£Z?  in  the  former  its  value  deduced  from  the 
latter  equation,  and  we  get 

(a  cos  B  +  b  cos  A)  dA  =  -  a  cos  B  dO; 

or  c  dA  =  -  a  cos  B  dC,  as  before. 

no.  Equation  connecting  the  Variations  of  two 
Sides  and  the  opposite  Angles. — In  general,  if  we  take 
the  logarithmic  differential  of  the  equation 

a  sin  B  =  b  sin  A, 

regarding  a,  b,  A,  B,  as  variables,  we  get 

da         dB       db       dA  . 

a      tan  B      b      tan  A'  * 

in.  I^anden's  Transformation. — The  result  in  equa- 
tion (16)  admits  of  being  transformed  into 

dA     __dOm 

a  cos  B  c 

but 


c  =  */a?  +  b%  -  20b  cos  C,  and  a  cos  B  =  */a%  -  b*  sin2^l; 

hence  we  get 

dA  dO 


*s/a*  -  ¥  sin  %A        \/c?  +  bz  -  zab  cos  C* 


Examples  in  Spherical  Trigonometry.  133 

If  C  be  denoted  by  1800  -  20i,  the  angle  at  A  by  <£,  and 

b 

-  by  h,  the  preceding  equation  becomes 

a 

d(J>  2d(pi  2d(f)i 


*y 1  -  ¥  sin2 $      \/ \  +  2kco8  2<f>i  +  k2      ^/{i  +k)2-  4&sin2$i 

2  *fyi 

~  (1  +  &)  a/ 1  -  ^i2  sin3  0i '  ^  ' 

where  Id  =        , . 

1  +  h 

Also,  the  equation  a  sin  5  =  5  sin  ^  becomes 

sin  (20!  -  (j))  =  h  sin  $. 

The  result  just  established  furnishes  a  proof  of  Landen's* 
transformation  in  Elliptic  Functions. 

We  shall  next  investigate  some  analogous  formulse  in 
Spherical  Trigonometry. 

112.  Relation  connecting  the  Variations  of  Three 
Sides  and  One  Angle. — Differentiating  the  well-known 
relation 

cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  C, 
regarding  a  and  b  as  constants,  we  get 
dc      sin  a  sin  b  sin  C 


dC  sin  c 


sin  a  sin  B. 


dc 
Again,  the  value  of  — ,  when  b  and  C  are  constants,  can 

be  easily  determined  geometrically  as  follows  :— 


•  This  transformation  is  often  attributed  to  Lagrange ;  it  had,  however,  been 
previously  arrived  at  by  Landen.  (See  Philosophical  Transactions,  177  i  and 
17750 


*34  Partial  Differentiation, 

In  the  spherical  triangle  ABO,  making  a  construction 
similar  to  that  of  Fig.  4,  Art.  108,  we  have 

^^  =  A#;  /.  -7-  =  limit  01  — -  =  -=r=^ 

(in  the  limit)  =  cos  B. 
Similarly,  when  a  and   C  are  con- 
stants. —  =  cos  A. 
do 

Hence,  finally,  lg*   ' 

de  =  cos  B  da  +  cos  Adb  +  sin  0  sin  i?  <#(7.  (19) 

This  result  can  also  be  obtained  by  a  process  of  diffe- 
rentiation. This  method  is  left  as  an  exercise  for  the 
student. 

As,  in  the  corresponding  case    of  plane  triangles,  we 

have  assumed  that  AB'  =  AB  in  the  limit ;    i.e.,   that 

AB'  —  AD 

— — —  is  infinitely  small  in  comparison  with  AD  in  the 

limit ;  this  assumption  may  be  stated  otherwise,  thus : — 
If  the  angle  A  of  a  right-angled  spherical  triangle  be 

very  small,  then  the  ratio  — — -   becomes  very  small  at  the 

j± 

same  time,  where  e  and  b  have  their  usual  significations. 

This  result  is  easily  established,  for  by  Napier's  rules  we 

have 

tan  b      sin  b  cos  c 

cos  A  = = ; — ; —  ; 

tan  c      cos  0  sin  c 

1  -  cos  A     sin  c  cos  b  -  cos  c  sin  b     sin  (c  -  b)  \ 


or 

sin 


1  +  cos  A     sin  c  cos  b  +  cos  c  sin  b     sin  (c  +  b)' 


,     o-A    •    /       7\         sm  (c-b)      .    ,      ,v,     A 
(c-b)  =  tan2  —  sm  (c+b);  .*.       v       '  =  sin(c  +  o)tan— . 

tan  — 

<;  2 


But  the  right-hand  side  of  this  equation  becomes  very  small 
along  with  A,  and  consequently  c-b  becomes  at  the  same 
time  very  small  in  comparison  with  that  angle. 


Examples  in  Spherical  Trigonometry.  135 

The  formula  (19)  can  also  be  written  in  the  form 

7^  dc  da  db 

dC  = = — ^  - 7 5  -    •    7  .       r  (20) 

sm  a  sm  B     sin  a  tan  i*      sin  0  tan  ^4  x     ' 

The  corresponding  formulae  for  the  differentials  of  A  and  B 
are  obtained  by  an  interchange  of  letters. 

Again,  from  any  equation  in  Spherical  Trigonometry 
another  can  be  derived  by  aid  of  the  polar  triangle. 

Thus,  by  this  transformation,  formula  (19)  becomes 

dO  =  -  cos  b  dA  -  cos  a  dB  +  sin  A  sin  b  dc.  (2 1 ) 

These,  and  the  analogous  f  ormulae,  are  of  importance  in 
Astronomy  in  determining  the  errors  in  a  computed  angular 
distance  arising  from  small  errors  in  observation.  They  also 
enable  us  to  determine  the  most  favourable  positions  for 
making  certain  observations ;  viz.,  those  in  which  small  errors 
in  observation  produce  the  least  error  in  the  required  result. 

113.  Remarks  on  Partial  Differentials. — The  be- 
ginner must  be  careful  to  attach  their  proper  significations  to 

the  expressions  — ,  — ,  &c,  in  each  case.     Thus  when  a  and 
aa    cl\y 

dc 
0  are  constants,  we  have  -^  =  sin  a  sin  B ;  but  when  A  and  a 

are  constants,  we  have  -77^  =  ■: ^:  these  are  quite  different 

dC     tanC  x 

dc 
quantities  represented  by  the  same  expression  — -. 

The  reason  is,  that  in  the  former  case  we  investigate  the 
ultimate  ratio  of  the  simultaneous  increments  of  a  side  and 
its  opposite  angle,  when  the  other  two  sides  are  considered  as 
constant ;  while  in  the  latter  we  investigate  the  similar  ratio 
when  one  side  and  its  opposite  angle  are  constant. 

Similar  remarks  apply  in  all  cases  of  partial  differentia- 
tion. 

When  our  formulae  are  applied  to  the  case  of  small  errors 
in  the  sides  and  angles  of  a  triangle,  it  is  usual  to  designate 
these  errors  by  Aa,  Ab,  Ac,  A  A,  AB,  AC;  and  when  these 
expressions  are  substituted  for  da,  db,  &c,  in  our  formulae, 
they  give  approximate  results. 


136  Partial  Differentiation. 

For  instance  (19)  becomes  in  this  case 

Ac  =  Aa  cos  B  +  Ab  cos  A  +  AC  sin  a  sin  B  ;         (22) 

and  similarly  in  other  cases. 

It  is  easily  seen  that  the  error  arising  in  the  application  of 
these  formulae  to  sucli  cases  is  a  small  quantity  of  the  second 
order ;  that  is,  it  involves  the  squares  and  products  of  the 
small  quantities  Aa,  Ab,  Ac,  &c.  This  will  also  appear  more 
fully  from  the  results  arrived  at  in  a  subsequent  chapter. 

114.  Theorem. — If  the  base  c,  and  the  vertical  angle  C, 
of  a  spherical  triangle  be  constant,  formula  (19)  becomes 

da  db 

+ =.  =  o. 


cos  A     cos  B 

Now,  writing  $  instead  of  a,  \p  instead  of  b,  and  k  for 

—. — ,  this  equation  becomes 
sin  c 


.        1      sin  ^4 

since  k  -  -= 

\                 Bina 

sin^X 
sin  b  J 

d(p 

1 

dip 

\/ 1  -  k2  sin20  '  */ 1  -  W  sin2i// 
where  $  and  \p  are  connected  by  the  following*  relation : — 
cos  c  =  cos  $  cos  \p  +  sin  <p  sin  xp  cos  C, 


or         cos  c  =  cos  0  cos  \p  +  sin  0  sin  \p  \/i  -  k2  sin2  c. 
115.  In  a  Spherical  Triangle,  to  prove  that 

da  db  dc  .     v 

+  nrB  +  rrr77  =  °J  (24) 


cos  A      cos  B     cos  C 


when  — : is  constant. 

sin  c 


*  This  mode  of  establishing  the  connexion  between  Elliptic  Functions  by 
aid  of  Spherical  Trigonometry  is  due  to  Lagrange. 


Examples  in  Spherical  Trigonometry.  137 

Let  sin  0  =  k  sin  c,  and  we  get 

„     k  cos  c  7       sin  A  cos  c  _ 

dC  =  - yz-dc  = fydc: 

cos  (7  sm  a  cos  0 

substitute  this  value  for  dC  in  (19),  and  it  becomes 

.    _  ^  7       cos  c  sin  -4  sin  B  _ 

cfc  =  cos  -4  ^0  +  cos  B  da  + 7= dc  ; 

cos  C 

.  __  _  _       /      cos  c  sin  -4  sin  2?\  7 

or        cos  .4  $0  +  cos  B  da  =    1 -pz dc 

\  cos  C         J 

cos  A  cosB  7 
= yr—  dc  ; 

cos  G7 

since  sin  A  sin  I?  cos  c  =  cos  (7  +  cos  A  cos  5. 

_  da  db  dc 

Hence  7  +  ■ ^  + y,  =  o. 

cos  A      cos  i?     cos  V 

Again,  since  cos  A  =  \/i  -  sin2 A  =  v  1  -  A2  sin2#,  &c, 
the  preceding  result  may  be  written  in  the  form 

da  db  dc 

yi-Fsin2a  +  vA-^sin2^  +  V^-^sin^  =  °'       ^ 

where  a,  b,  c,  are  connected  by  the  equation 

cos  c  =  cos  a  cos  b  +  sin  #  sin  b  yi  -  k?  sin2c. 

116.  Theorem  of  IJegendre. — "We  get  from  (24) 

cos  B  cos  Cda  +  cos  A  cos  (7d$  +  cos  B  cos  -4dc  =  o, 

or  (cos  A  -  sin  B  sin  (7  cos  a)  da  +  (cos  i?  -  sin  A  sin  O  cos  b)  db 

+  (cos  0  -  sin  ^  sin  B  cos  c)  <#c  =  o ; 

.*.  cos  Ada  +  cos  Bdb  +  cos  Ode 

=  sini?  sin  C^(sin  a)  +  sin^t  sin  (7<#  (sin  b)  +  sin  .4  sin  Bd  (sin  c) 

=  F  { sin  5  sin  cd  (sin  &)  +  sin  a  sin  c<#  (sin  5)  +  sin  a  sin  5rf  (sin  c) } 

=  k?d  (sin  a  sin  5  sin  c)  ; 


138  Partial  Differentiation. 

or     ^/i  -  k2  sin2  a  da  +  */\  -  k%  sin2  bdb  +  */\  -  k2  sin2  c  dc 

=  k2d  (sin  a  sin  &  sin  c) .  (26) 

This  furnishes  a  proof  of  Legendre's  formula  for  the  compa- 
rison of  Elliptic  Functions  of  the  second  species. 

The  most  important  application  of  these  results  has  place 
when  one  of  the  angles,  G  suppose,  is  obtuse ;  in  this  case 
cos  C  is  negative,  and  formula  (25)  becomes 

da  db  •  dc 

+ 


vA  -  k2  sin2#     \/ 1  -  k2  sin2  b     ^/i  -  k2  sinV 
where  the  relation  connecting  a,  b,  c  is 


cos  c  =  cos  a  cos  b  -  sin  a  sin  b  */i  -  k2  sin2c. 
In  like  manner,  equation  (26)  becomes,  in  this  case, 

*/ 1  -  k2  sin2  a  da  +  ^/i  -  k2  sin2  b  db 

=  */ 1  -  k2  sin2  cdc  +  k2  d  (sin  a  sin  b  sin  c) . 

117.  If  u  =  (p(%  +  at,  y  +  fit),  where  x,  y,  a,  fi,  are  in- 
dependent of  t,  and  of  each  other,  to  prove  that 

du        du     ~du  .     » 

db        dx        dy 

Let  x  =  x  +  at,    yf  =  y  +  fit; 

then  u  -  <p  (x',  yf), 

dxr  dy'  dx'  dyf      ~ 

and  a"1'*"1'  dt~a>W  =  li- 

Also,  since  y  is  independent  of  x,  we  have 

du      du  dx'      du         A    du      du 
dx     dx' dx       dx"  dy      dyT 

TT  du     du  dx'     du  dy'        du      ~  du 

dt      dx'  dt      dy'  dt         dx         dy 


Partial  Differentiation.  139 

In  like  manner,  if  x',  y',  z',  be  substituted  for  x  +  at,  y  +  fit, 
z  +  yt,  in  the  equation 

u  =  <p(x  +  at,  y  +  fit,  z  +  yt), 
it  becomes        u  =  $  {x,  y',  z') ; 

du     du  dx      du  dtf     du  dz 
dx     dx  dx      dy'  dx      dz' dx  9 

dx'  dy'  dz' 

dx        '  dx        9  dx 

du     du  du      du     du     du 

dx     dx'9  dy     dy'9   dz      dzr 

.      .  du     du  dx'     du  dy'     du  dz' 

gam       ~di~dx'~dt+dy'~di'{'  dz'H' 

,    ,  dx'  dy'  dz' 

but  !i  =  a>  W  =  /3'  Mmf' 

t-t  du        du      ^du         du  .     x 

Hence      _-a_  +  0_  +  y_  (28) 

tt't'  Uw  W-w  Cf* 

This  result  can  be  easily  extended  to  any  numberof  variables. 


140  Examples. 


Examples. 


•   -,  (%\      .   ,  fy\  j,  L  ,  d*  &y 

1.  If  u  =  sin"1  (  -  )  +  sir1  (  7- ) ,  prove  that  «w  =     ,  +  — . 

W  W  A/a2 -a;2      ^/b^-y2 

3.  Find  the  conditions  that  «,  a  function  of  a?,  2/,  2,  should  be  a  function  of 
a?  +  y  +  s. 

.        du     du      du 
Am.   — -  =  —  =  — . 
dx      dy     dz 

4.  If  f(ax  +  3y)  =  e,  find  ^.  „    -  |. 

5.  If /(w)  =  <p(v),  where  w  and  0  are  each  functions  of  #  and  y,  prove  that 

du  dv      dv  du 
dx  dy      dx  dy' 

-n.    1   1        t  „     du        du      . 

b.  Find  the  values  of  x  —  +-  y  — ,  -when 

(a)    m  =  — r- -, 

tw*  +  nyl 

03)   «  =  tan-i(^^V. 

7.  If  w  =  sin  ax  +  sin  fo/  +  tan-1  I  -  J ,  prove  that 

r    ,      zdy  -  yd* 

du  =  a  cos  ax  dx  +  0  cos  by  dy  A 5 =-. 

y2  +  z2 

n    ,du       -du  .      du  1  du        —logo; 

8.  If  u  =  logyx,    find  —  and  — .  Am,  —  =  — ,     —  =  — - — 5-—. 

dx         dy  dx     xlogy      dy      y  (log  yy 

q.  If  9  =  tan-1  — ,  prove  that 

y 

(x2  +  y2)  dd  =  ydx  —  xdy. 
10.  If  u  =  */*%  prove  that 

<?«  =  yxz-i  (xzdy  +  yz  log  ydx  4-  #y  l°g  ^z)« 


Examples.  141 

<Wa2-y2 


11.  If  a  +  y  a?  -  V2  —  Ve      a      j  prove  that 

dy_       -y 
dx     </'a2  -  y%' 

12.  In  a  spherical  triangle,  when  a,  b  are  constant,  prove  that 

dA     tan  A       ,  dO  sin  C 

-,  and  -^  =  — 


dB      tan  B  dB        sin  B  cos  .4 

13.  In  a  plane  triangle,  if  the  angles  and  sides  receive  small  variations, 
prove  that 

cAB  +  b  cos  A  AC  =  o  ;     a,  b  heing  constant, 

cos  CAb  +  cos  BAc  =  o  ;     a,  A  being  constant, 

ta,nAAb  =  bAG;  a,  B  being  constant. 

14.  The  base  c  of  a  spherical  triangle  is  measured,  and  the  two  adjacent 
base  angles  A}  B  are  found  by  observation.  Suppose  that  small  errors  dA,  dB 
are  committed  in  the  observations  of  A  and  B  ;  show  that  the  corresponding 
error  in  the  computed  value  of  C  is 

—  cos  adB  -  cos  bdA. 

15.  If  the  base  c  and  the  area  of  a  spherical  triangle  be  given,  prove  that 

a  b 

sm2  -dB  +  sin2  -dA  =  o. 

2  2 

16.  Given  the  base  and  the  vertical  angle  of  a  spherical  triangle,  prove  that 
the  variation  of  the  perpendicular  p  is  connected  with  the  variations  of  the  sides 
by  the  relation 

sin  Cdp  =  sin  s'da  +  sin  sdb, 

s  and  /  being  the  segments  into  which  the  perpendicular  divides  the  vertical 
angle. 

17.  In  a  plane  triangle,  if  the  sides  a,  b  be  constant,  prove  that  the  variations 
of  its  base  angles  are  connected  by  the  equation 

dA  dB 


y/cP-FsLtfA      */ '&  -  a?  sin2^' 

18.  Prove  the  following  relation  between  the  small  increments  in  two  sides 
and  the  opposite  angles  of  a  spherical  triangle, 

da         dB         dA  db 

+  : ^  =  : .  + 


tan  a     tan  B     tan  A     tan  b' 

19.  In  a  right-angled  spherical  triangle,  prove  that,  if  A  be  invariable 
sin  2cdb  =  sin  2bdc ;  and  if  c  be  invariable,  tan  add  +  tan  bdb  =  o. 


142  Examples. 

20.  If  a  be  one  of  the  equal  sides  of  an  isosceles  spherical  triangle,  whose 
vertical  angle  is  very  small,  and  represented  by  dec,  prove  that  the  quantity  by 

■which  either  base  angle  falls  short  of  a  right  angle  is  -  cos  a  da. 

21.  In  a  spherical  triangle,  if  one  angle  C  be  given,  as  well  as  the  sum  of 
the  other  angles,  prove  that 

da         db 

+  -r—y  =  O. 


sin  a      sin  b 
22.  If  all  the  parts  of  a  spherical  triangle  vary,  then  will 

cos  Ada  +  cos  Bdb  +  cos  Cde  =  Jed  {h  sin  a  sin  b  sin  c) ; 
sin  A      sin  B      sin  0 


where  #  = 


sin  a      sin  b       sin  c ' 


da           db           dc  ii      ■».-    «,/i\ 

Also ~  + -H -  =  tan-4tan2?tanC^I  -  J . 


cos  A      cos  B      cos  C 


These  theorems  can  be  transformed  by  aid  of  the  polar  triangle  ? — M i  Cullagh, 
Fellowship  Examination,  1837. 

These  are  more  general  than  the  theorems  contained  in  Arts.  115  and  116, 
and  can  be  deduced  by  the  same  method  without  difficulty. 

23.  If  z  =  ty  (x2  —  y2),  prove  that 

dz         dz 
dx      -  dy 


24.  If  z  =  -/  f  -  ] ,  prove  that 
x     \xj 


dz         dz 
x-v  +  2/T  +2  =  0. 
dx        dy 

25.  Find  —  and  —   when  x,  y,  z  are  connected  by  two  equations  of  the 

CtCC  CtfOO 

form 

/(#,*/,  *)=o,         <p(x,y,z)=o. 

df  d<p      df  d(j> 
dy      dx  dz       dz  dx 
dx  "  dfd$  _df_dj>' 
dz  dy      dy  dz 

df  dip  df  d<p 

dz  _  dy  dx  dx  dy 

"     ~dx  ~  dfd(f>  df  d<p 

dz  dy  dy  dz 


Examples.  1 43 

26.  Prove  that  any  root  of  the  following  equation  in  y, 

ym  +  xy  =  1, 
satisfies  the  differential  equation 

„  d2y      ,  dy3       .  x     dy2 

37.  How  can  we  ascertain  whether  an  expression  such  as 

cj>(i%,y)+*/-ity{x,y) 
admits  of  being  reduced  to  the  form 

d<p      dty        d(p         dfy 
dx      dy'        dy  dx 

28.  If  IX  +  mY+  nZ,  I'X  +  m'Y  +  n'Z,  l"X  +  m"Y  +  n"Z,  be  substituted 
for  x,  y,  z,  in  the  quadratic  expression  of  Art.  107  ;  and  if  a',  b',  c',  d't  e',/',  be 
the  respective  coefficients  in  the  new  expression,  prove  that 


a!    f     e' 

f  v  a 

e'     d'    c' 


I     a    f    e 
f    b    d    |=o. 
e    d    e    I 


29.  If  the  transformation  be  orthogonal,  i.  e.  if  #3  +  y2  +  z2  =  X2  +  Y2  +  Z\ 
prove  that  the  preceding  determinants  are  equal  to  one  another. 

29.  If  u  be  a  function  of  £,  77,  £  and  £  =  «/+-,   77  =  z  +  -,    £  =  #  +  -^ 

2  x  y 

show  that 

<?m        du        dn        du        du       .du         I    du        du  du\ 

*fo        %         «fo         dl-         drj        d£         \    d£        d%  dijf 


(     144     ) 


CHAPTEE  VI. 

SUCCESSIVE   DIFFERENTIATION  OF  FUNCTIONS   OF  TWO  OH  MORE 

VARIABLES. 

1 1 8.  Successive  Partial  Differ eiitiatioii. — We  have  in 
the  preceding  chapter  considered  the  manner  of  determining 
the  partial  differential  coefficients  of  the  first  order  in  a  func- 
tion of  any  number  of  variables. 

If  u  be  a  function  of  x,  y,  z,  &c.,  the  expression 

du     du      du    . 
dx>     ~dy>     dz~>     C'' 

being  also  functions  x,  y,  z9  &c,  admit  of  being  differen- 
tiated in  the  same  manner  as  the  original  function ;  and  the 

partial  differential  coefficient  of  — ,  when  x  alone  varies,  is 
■"■  dx 

denoted  by 

d  fdu\      d2u 


dx  \dxf      dx29 

as  in  the  case  of  a  single  variable. 

du 
Similarly,  the  partial  differential  coefficient  of  — ,  when  y 

ttX 

alone  varies,  is  represented  by 

d  fdu\        d2u 
dy  \dxj        dydx9 

and,  in  general,  - — —  denotes  that  the  function  u  is  first 
°  dymdxn 

differentiated  n  times  in  succession,  supposing  x  alone  to 

vary,  and  the  resulting  function  afterwards  differentiated  m 

times  in  succession,  where  y  alone  is  supposed  to  vary ;  and 

similarly  in  all  other  cases. 


The  Order  of  Differentiation  is  Indifferent.  145 

We  now  proceed  to  show  that  the  values  of  these  partial 
derived  functions  are  independent  of  the  order  in  which  the 
variables  are  supposed  to  change. 

119.  If  u  foe  a  Function  ofx  and  y,  to  prove  that 

d  (du\      d  fdu\         d2u       d2u 
dy\dos)     dx\dyj        dydx     dxdy*  ' 

where  x  and  y  are  independent  of  each  other. 

du 
Let  u  =  §  (x,  y),  then  —  represents  the  limiting  value  of 

cix 

<p(x  +  h,y)-<j>  (x,  y) 
h 

when  h  is  infinitely  small. 

This  expression  being  regarded  as  a  function  of  y,  let  y 

become  y  +%  x  remaining  constant ;  then  -f-(-r)  is  the 
limiting  value  of 

(ft  (x  +  h,  y  +  h)  -  (p  (x,  y  +  k)  -  <p  (x  +  h,  y)  +  <j>  (x,  y) 

hk 

when  both  h  and  k  become  infinitely  small,  or  evanescent, 

Q/tl 

In  like  manner  —  is  the  limiting  value  of 

<j>(x,y  +  h)  -(j>(x,y) 
k 

d  1  din  \ 

when  k  is  infinitely  small ;  hence  —  (  — )  is  the  limiting  value 

of 

$  (x  +  h,  y  +  k)  -  <j>  (x  +  h,  y)  -  ^  (x,  y  +  k)  +  <j>  (x,  y) 

hk 

when  both  h  and  k  are  infinitely  small. 

Since  this  function  is  the  same  as  the  preceding  for  all 

L 


146 


Successive  Partial  Differentiation. 


finite  values  of  h  and  k,  it  will  continue  to  be  so  in  the  limit; 
hence  we  have 

d  fdu\      d  fdu\ 

dec  \dyj      dy  \dxj 


In  like  manner 


for  by  the  preceding 


dhi 
dx2dy 

d2u 
dxdy      dydx ' 


d3u 
dydx2  9 
d2u 


d  f  dhi 


dx  \dxdy 


d_ 

dx 


d2u 

d     d 

du 

d     d 

du 

dydx 

dx'  dy 

dx 

dy'  dx 

dx 

similarly  in  all  other  cases.     Hence,  in  general, 

d?**u        cP*eu 
dxpdyq      dyqdxp' 

Again,  in  the  case  of  functions  of  three  or  more  variables, 
by  similar  reasoning  it  can  be  proved  that 

d3u  d3u      . 

dzdxdy     dxdydz' 

Hence  we  infer  that  the  order  of  differentiation  is  in  all  cases 
indifferent,  provided  the  variables  are  independent  of  each 
other. 


1.  If  u  =  d>[  -  J, 

2.  If  u  =  tan-1 1  -  J , 

\yl 

3.  If  u  =  sin  (axn  +  byn), 


Examples  toe  Verification. 

d?u 


verify  that 


dydx 

d3u 
dy2dx 

d^u 


d2u 

dxdy 

dzu 


dxdy*1' 
d*u 


dx^dy2,      dy2dxz' 


120.  Condition  that  Pdx  +  Qdy  shall  he  a  total 
differential. — This  implies  that  P  dx  +  Q  dy  should  be  the 
exact  differential  of  some  function  of  x  and  y.  Denoting  this 
function  by  u,  then 

du  =  P  dx  +  Q  dy, 


(2) 


Condition  for  a  Total  Differential  147 

and,  by  (1),  Art.  95,  we  must  have 

_     du  ~      du 

P  =  — ,         Q  =  —  ; 

ax  ay 

dP      d2u       dQ      d2u 
rfy      %^'     dx      dxdy 

Hence  the  required  condition  is 

dPdQ 
dp       dx 

121.  If  u  be  any  Function  of  x  and  ?/,  to  prove  that 

i(*<">SK(™S).        (3) 

where  x  and  ^  are  independent  variables. 

Here  each  side,  on  differentiation,  becomes 

__ .  v   a  u       Tit  /  \      an  0 

dxdy  ' dxdy' 

122.  more  generally,  to  prove  that 

d  f  dv\      d  f  dv\ 

dy  \  dx)      dx  \  dy/  *  ' 

where  u  and  v  are  both  functions  of  z,  and  z  is  a  function  of 
x  and  y. 

._  <£  /  cfo\      dud/o  d2v 

For  -T-    w-r-    =  -T-  t-  +  w 


but 


c?w     e?w  dz      dv      dv  dz 
dy     dz  dy*     dx     dz  dx  ' 

d  (  dv\      du  dv  dz  dz  d2v 

—     U—     =-——-  —  —   +  u 


dy  \  dx)     dz  dz  dx  dy        dydx ' 

and  —  ( u—  )  has  evidently  the  same  value. 

dx\  dy }  J 


l  2 


148  Successive  Partial  Differentiation. 

123.  Euler's   Theorem    of  Homogeneous    Func- 
tions.— In  Art.  102  it  has  been  shown  that 

du        du 

x  —  +  y  —  =  nu, 
dx        dy 

where  u  is  a  homogeneous  function  of  the  nth   degree  in 
v  and  y. 

Moreover,  as  —  and  —  are  homogeneous  functions  of  the 

dx         dy 

degree  n  -  1,  we  have,  by  the  same  theorem, 


dx 


x 


d  fdu\  d  fdu\  .           du 

Ix  \clxj  dy  \dxj  '  dx* 

d  fdu\  d  fdu\  .du^ 

dx  \dyj  dy  \dyj  dy' 


multiplying  the  former  of  these  equations  by  ss,  and  the 
latter  by  y,  we  get,  after  addition, 


„  d2u  d2u        „  d2u 


,         s  (  du        du\ 

x*  -—  +  2xy  -7—7 -  +  y'  -=-=  =  {n-i)[a;—  +  y—\ 

dx2  y  dxdy     *  dy2     v        ;  \  dx     *  dy) 

=  (n-i)nu.  (5) 

This  result  can  be  readily  extended  to  homogeneous 
functions  of  any  number  of  independent  variables. 

A  more  complete  investigation  of  Euler's  Theorems  will 
be  found  in  Chapter  VIII. 

124.  To  find  the  Successive  Differential  Coeffi- 
cients with  respect  to  t9  of  tbe  Function 

(j>(x  +  at,    y  +  fit), 

where  x,  y,  a,  fi,  are  independent  of  t,  and  of  each  other. 

By  Ait.  1 1 7  we  have  in  this  case,  where  0  stands  for  the 
expression  <p(x  +  at,  y  +  fit), 

d<})        d<f>      _  d(j> 
dt        dx         dy 


Differentiation  qf(j>(x  +  at,  y  +  j3£).  149 

Hence  g-;»(?Ws(£) 

dtf        dt\dx)         dt\dy) 

d  fd(f\     R  d  (d<$\ 
~adx\dtJ     P  dy\dt) 

dx  (   dx         dy)         dy  {   dx         dy) 

=  a^  +  2aR  £*.  +  &*¥$.  (6) 

dx2  dxdy         dy2'  ' 

This  result  can  also  "be  written  in  the  form 

d2<p      (    d  d)  d$      id      a  d)2  .  x 

dP  =  \aTx  +  iiTy)Tt=\aTx  +  Ply\*>  (7) 

f    d  d\2 

in  which  f  a  —  +  j3  — )  is  supposed  to  be  developed  in  the 

usual  manner,  and  -7-f ,  &e.,  substituted  for  f  —  J  0,  &c. 

d?<$> 
Again,  to  find  — . 

ccz 

dtz  ~  dt\df)~  dt\adx+"dj/J* 

dx         dy)  dt      \  dx     '    dy)  \   dx     ^  dy) 


\  dx     ^  dy)  ^' 

By  induction  from  the  preceding  it  can  be  readily  shown 
that 

dn<p      (    d      „d\n 

W=\arx  +  Pdy)*' 

This  expression,  when  expanded  by  the  Binomial  Theorem, 
gives  the  nth  differential  coefficient  of  the  function  in  terms  of 
its  partial  differential  coefficients  of  the  nth  order  in  x  and  y. 


150 


Examples. 


Examples. 


d2u         dru 


1.  If  u  =  sin  (x2y),  verify  the  equation 

2.  If  u  =  sin  (2/  +  ##)  +  (y  —  ax)2,  prove  that 

d2u  d2u 
—  =  a2  — . 
dx2         dy2 

3.  In  general,  if  u  -f{y  +  ax)  +  $  (y  —  ax),  prove  that 


d2u 
dx2 


=  «2 


d2u 


dy 


z' 


4.  If  u  =  «/*,  prove  that 


dxdy 


y^ii  +  xlogy)  = 


d2u 
dydx 


5.  If  u  = 


xyz 


ax  +  by  +  es 


,  find  the  values  of 


^%     d2w  d2w 

^2'     ^'   a       d?' 


6.  If  w  =  (#2  +  y2)*,  prove  that 


.^22£ 


+  2xy 


d2u 


+  y2 


d?u 


dx2  '     ~*  dxdy  '  ff  dy2 
7.  If  u  =  (#3  +  2/3)»,  prove  that 


o. 


.  d2u 


+  2#y- 


^2W 


+ 


2^-3 


8.  If  V=Ayd  +  $By2x  +  $Cyx2  +  Dx*,  prove  that 

<PVd_n  _      fV_dVdV     d2VdV2_      v 
dx2  dy2  dxdy  dx  dy       dy2  dx2 


%2>  -  «y,  y% 

A,  B,    G 

B,  C,    D 


and  show  that  the  left-hand  side  of  this  equation  vanishes  -when  V  is  a  perfect 
cube. 


9.  liu  = 


(s2  +  y%  +  z2)i 


i?  prove  that 


dht      d2u      dhi 

1 -j =  o. 

dx2      dy2      dz2 


(     i5i     ) 


CHAPTEB  VII. 

lagrange's  theorem. 

125.  Lagrange's  Theorem. — Suppose  that  we  are  given 
the  equation 

z  =  x+.y<j>(z),  (1) 

in  which  x  and  y  are  independent  variables,  and  it  is  required 
to  expand  any  function  of  z  in  ascending  powers  of  y. 

Let  the  function  be  denoted  by  F(z),  or  by  u,  and,  by 
Maclaurin's  theorem,  we  have 

.  y  f^t\   j[_  (d%u\  yn  _  (d*u\  &    (y 

1  \dyJo     1  .  2  \dy2  Jo  1.2...  n\dyn  J0        "' 

(/Jf/\  nil 

—  ) ,  &c,  represent  the  values  of  u,  -7-,  &c,  when 
dyjo  .  dV 

zero  is  substituted  for  y  after  differentiation. 
It  is  evident  that  u0  =  F(x). 

To  find  the  other  terms,  we  get  by  differentiating  (1)  with 
respect  to  a?,  and  also  with  respect  to  y, 

dz  ,.  .  dz  dz         .  .  ,.  .dz 

.  dz         ,  .dz 

hence  —  =  6[z)-—. 

dy      r     dx 

Also,  since  u  is  a  function  of  z,  we  have 

c?w      ^w  dz         du      du  dz 
dx     dz  dx'       dy     dz  dy* 


152  Lagrange's  Theorem. 

hence  we  obtain 

du       f  jdu  ,  v 

Ty  m  *®&  (3) 

Again,  denoting  $(s)  by  Z,  we  have  by  Art.  121,  since 
Z  is  a  function  of  u, 


( „du\       d  ( r,du\      d2u   „         ,  N 


d2u       d  (      du\  .  v 

dyz      dx\      dx)' 

t-t  ,  d3u        dz   ( mdu\ 

Hence  also  -7-=  =  -=-=-  \"  t  > 

since  #  and  y  are  independent  variables ; 

«  i(*S-i(»S-£(**W» 

d2    (^  du\  =  fdV  (z%  du\  m 
dxdy  \      dx)      \dx)  \     dx) ' 

To  prove  that  the  law  here  indicated  is  general,  suppose 

that  *±J*X*(z&\l 

dyn      \dx)     \      dx) ' 

dy\     dx)      dx\      dy )      dx\        dx/ 

dn      f      du\  _  d?_  ( yn+1  du\  t 
dxn~x  dy\     dx)     dxn\        dx)' 

andheace  _=y^_].  (6) 


Lagrange's  Theorem. 


153 


This  shows  that  if  the  proposed  law  hold  for  any  integer 

n,  it  holds  for  the  integer  n  +  1 ;  but  it  has  been  found  to  hold 

for  n  =  2  and  n  =  3  ;  accordingly  it  holds  for  all  integral  values 

of  n. 

du    d  ?/ 
It  remains  to  find  the  values  of  — ,  -— ,   &c.   when  we 

ay    ay2 

make  y  =  o.  Since  on  this  hypothesis  Z  or  0(s)  becomes 
0(#),  and  -j-  becomes  J"  '  or  F'(x),  it  is  evident  from  (3), 
(4),  (5),  (6),  that  the  values  of 

du      d2u     d3u         dn™u 
dy9     dy19     dy3  ' 

become  at  the  same  time 


dy 


,«+!> 


#(*)  F'(x), 


L 

dx 


{*(*)*■*"(*) 


dx% 


{#(«.)  j 'j"(«) 


dn   ~ 

Consequently  formula  (2)  becomes 

r '  [<j>{x)Y F'  (x) 


y 


F(z)  =  F{x)  +  *-  <h(x)F\x)  +J?-4- 
1  N  '      1  .2  dx 


.  .  .  + 


y7l+l  fin 

i,2...(n+  1)  dxn 


[(j>(x)}n+1F'(x) 


+  &C. 


+  &c. 
(7) 


This  expansion  is  called  Lagrange's  Theorem. 

If  it  be  merely  required  to  expand  g,  we  get,  on  making 
F(z)=z, 

z  =  x+y-<i>(x)  +-£-4-  {0W}2+&c 
1  rw       i.2dx  XYy  n 


f 


1 .2  . .  ,n  dx' 


*,*{*(»)}*  +  **. 


(8) 


154  Laplace7 s  Theorem. 

126.  Laplace's  Theorem. — More  generally,  suppose 
that  we  are  given 

z=f{%  +  y<P(z)},  (9) 

and  that  it  is  required  to  expand  any  function  F(z)  in  ascend- 
ing powers  of  y. 

Let  t  =  x  +  y<j)(z),  then  z  =f(t),  and  we  have 

*  =  *  +  ?*{/(*)}■  (10) 

Also  F(z)  =  F {/(£)}  ;  and  the  question  reduces  to  the 
expansion  of  the  function  F  {/(£)}  in  ascending  powers  of  y 
by  aid  of  (10) ;  accordingly,  formula  (7)  becomes  in  this  case 


*(.)  =  F {f(t)\  -F  {/[»))  +  7  *(/(*))  *" {/Ml  +  &°- 


1  . 2  .  . .  (w  +  1 )  dx% 


This  formula  is  called  Laplace's  Theorem,  and  is,  as  we 
have  seen,  an  immediate  deduction  from  the  Theorem  of 
Lagrange.  These  theorems  evidently  only  hold  when  the 
expansions  are  convergent  series. 


Examples.  155 

Examples. 

1.  Expand  2,  being  given  the  equation 

z  =  a  +  bz3. 

Here  x  =  a,  y  =  b,  cp{z)  =  23, 

and  "we  get,  from  formula  (8), 

z  =  a  +  ba3  +  3b2aP  +  i2b3a7  +  &c. 

Lagrange  has  shown  that  this  expansion  represents  the  least  root  of  the  pro- 
posed cuhic,  and  that  a  similar  principle  holds  in  like  cases. 

2.  Given  z  =  a  +  bzn,  find  the  expansion  of  z. 

b*                                       b2 
Ans.    z  =  a  +  anb  4-  inaln~x f-  im[v>i  —  i)  a3n~2 1-  &c. 

I   .  2  I.2.3 

3.  Given  z  =  %  +  ye",  find  the  expansion  of  z. 

Ans.  z=x  +  yex  +  y2e2x  +  -z—  xe3x  +  — - —  <&eix  +  &c. 

1.2  1.2.3 

4.  z  —  a  +  e  sin  2,  expand  (1)  2,  (2)  sin  2. 

e*      d  e3       f  d\2 

(1).  -4ws.  2  =  0  +  0  sin  #  + —  (sin2#)  + 1  —  1  (sin%)  +  &c. 

1  .  2  da  1.2.3  \daj 

(2).      ,,     sin  2  =  sin  a  +  e  sin  a  cos  a  4 —  (sin5#  cos  a)  +  &c. 

1  .  2  da 

5.  If  2  =  a  +  -  (22  —  1),  prove  that 

2 

%{a2-  1)        #2     rf    /a2-  i\2 

z  =  «  +  -  v 1  + —  ' 

1       2  i  .  2  da 


+ 


1 .2  ...  n 
6.  Hence  prove  that 


. 2  . . .  w \«0/    \     2     / 


+ 
1 


(     156     ) 


CHAPTER  Yin. 

EXTENSION    OF     TAYLOR'S    THEOREM    TO    FUNCTIONS    OF    TWO 
OR  MORE   VARIABLES. 

127.  Expansion  of  $  (x  +  h,  y  +  k) .  Suppose  u  to  be  a  func- 
tion of  two  variables  x  and  y,  represented  by  the  equation 
u  =  ${x,  y) ;  then  substituting  x  +  h  for  x,  we  get,  by  Taylor's 
Theorem, 

d  h2    d2 

<f>(x  +  h,y)  =  ${x, y)  +  h—  (0 (x, y)}  +  —  —  {<j>(x,  y))  +  &c. 

Again,  let  y  become  y  +  k}  and  we  get 

d 
<$>(x  +  h,y  +  k)  =  (j>(x,  y  +  k)  +  h—  {^(x,  y  +  k)) 


h2     d2 

+  7r2^i^^2/+^)+&0,    (l) 


But 


d  k2    d2 

,  du       W    d2u      0 

-  u  +  k—  + —  +  &c. 

dy      1.2  ay* 

Also 

d  du  d2u       hk2    d3u       „ 

dx  W*  9*       ''  ~     dx  dxdy      1  . 2~dxdy2         "' 


and 


JL  *Lt  k\\-———  -^-      & 

1 .  2  ffe2  '^  '  ^       ' '  "  1  .  2  dx2      1.2  d#2dy 


Extension  of  Taylor' }s  Theorem.  157 

Substituting  these  values  in  (1),  we  get 
,  /       7  7\*  .  du     7du 

hz  d2u      7T    d2u        h%  d2u      0  .  . 

;i  +  M  =  +  -i:3  +  &c-     (2) 


1 . 2  dx%  dxdy      1 .  2  dy1 

128.  This  expansion  can  also  be  arrived  at  otherwise  as 
follows : — Substitute  x  +  at  and  y  +  fit  for  x  andy,  respectively, 
in  the  expression  $  (x,  y),  then  the  new  function 

$  (x  +  at,  y  +  fit), 

in  which  x,  y,  a,  fi,  are  constants  with  respect  to  t,  may  be 
regarded  as  a  function  of  t,  and  represented  by  F(t) ;  thus 

<p  (x  +  at,  y  +  fit)  =  .F(tf). 

The  latter  function  F(t),   when  expanded  by  Maclaurin's 
Theorem,  becomes,  by  Art.  79, 

F{()=F{p)  +  t-F\o)+^-t"{o)  +  .., 

+  k*wW,  (3) 


where  F'(o)  is  the  value  of  F{t)  when  £  =  o,  i.  e.  F(o)  =  #  (#,  y) 
=  w;  also  -F'(o),  F"{p),  &c.  are  the  values  of 

d(j>     d2<f>     x 
It9     ~a¥9       C,, 

when  t  =  o  ;  where  0  stands  for  (f>(x  +  at,  y  +  fit). 
Moreover,  by  Art.  1 1 7,  we  have 

d(f>        d<j)      3  d$ 
dt        dx        dy9 


*  Since  it  is  indifferent  whether  we  first  change  x  into  %  +  h,  and  afterwards 
change  y  into  y  4-  Jc,  or  vice  versa  ;  the  expansion  given  above  furnishes  an  in- 
dependent proof  of  the  results  arrived  at  in  Art.  119. 


158  Extension  of  Taylor's  Theorem, 

but,  when  t  =  o,  $(%  +  at,  y  +  fit)  becomes  u,  or  F(o),  and  -J 

becomes  a  —  +  8  —  at  the  same  time. 
dx         ay 

_  _,.  .         du      0du 

Hence  F  (o)  =  ay  +  p—. 

Also,  by  the  same  Article, 

df  "     ^3  Pdxdy     p  dy* 

which,  when  £  =  o,  reduces  to 

&c.  &c.  &c. 

These  equations  may  also  be  written  in  the  symbolic 
form 


*wm-(«s+0|)"* 

Again,  f  a  —  j  u  =  ar  — ,  &c,  since  a,  |3,  are  independent 

of  x  and  ?/ :  and  hence  the  general  term  in  the  expansion  of 
F(t)  can  be  at  once  written  down  by  aid  of  the  Binomial 
Theorem. 


Extension  of  Taylor 's  Theorem.  159 

Finally,  we  have,  on  substituting  h  for  at,  and  k  for  fit, 


du     ,  du       h2    d2u  d2u 

dy      1.2  dx2  dxdy 


.  _.  _  au      .,  au        it     a  u 

d>(x  +  h,  y  +  k)  =u  +  h—  +  k  —  + _-;  +  hk 

r  v  *       '  dx        dy      1  .  2  dxr 


k2   d2u  1     /.  d      _  dy*1    .       a7  07\     /  \ 

+  iti^+-+!^t(^  +  ^J  *(*  +  «*»*  +  «)•  (5) 

129.  Expansion  of  (j>  (x  +  h,  y  +  k,  z  +  /). — A  function 
of  three  variables,  x,  y,  z,  admits  of  being  treated  in  a  similar 
manner,  and  accordingly  the  expression 

<j)(x  +  at,  y  +  fit,  %  +  yt), 

when  u  is  substituted  for  <j)(x,  y,  z),  becomes 


f    (    d     nd         d\2 
+  n\adi  +  Pdv  +  ydz)U  +  &0-' 


or 


<b(x  +  h.  y  +  k,  z  +  I)  =  u  +  [h—  +  k—-  +  l  —  )u 
r  x  \  dx        dy       dz) 

1     f7  d  d      7d\2 

1  .  2  \  e?#        ay       dzj 

du       du       du        h2    d2u       k2    d2u        P    d2u 
dx       dy      dz      1.2  dx2      1  .  2  d^2      1  .  2  dz2 

d2u  d2m  d2u 

+  hk——r  +  Ih  7-r  +  kl-r—r  +  &c.  (6) 

a##y         asafl?         ayws 

The  general  term  in  this  expansion,   and  also  the  re- 
mainder after  n  terms,  can  be  easily  written  down. 


1 60  Extension  of  Taylor's  Theorem. 

These  results  admit  of  obvious  generalization  for  any 
number  of  variables. 

Also,  by  making  x,  y,  %  eacb  cypher  in  (6),  we  have 


*  (a,  *o -M.+ *(!)+*(!) 


du\        fdu\ 
[dzjc 


—  f — ^ 

1  .  2  \dx2J0 


where  ( — ) ,  ( —  ] ,  .  .  .  denote  the  values  which  the  functions 
\dxJo  \dyJo 

df/    dm 

—,—,...  assume  on  making  x  =  o,  y  =  o,  and  %  =  o. 

dx  dy 

This    result    may    be    regarded    as    the    extension    of 

Maelaurin's  Theorem. 

130.  Symbolic  Expression  for  preceding  Results. — 

Since 


n±+k±  ( 1  d       _  d\        1     (.  d      _  d\* 

\   dx        dyj      1  .  2  \   dx        dy) 

1  ft  d       j  dX    x 

fs  \  dx        dy ) 

equation  (5)  may  be  written  in  the  shape 

ehdx^hTy^^  y^  ^^[x  +  hiy  +  k).  (7) 


This  is  analogous  to  the  form  given  for  Taylor's  Theorem 
in  Art.  67,  and  may  be  deduced  from  it  as  follows  : — 

d 

We  have  seen  that  the  operation  represented  by  eh<ke 
when  applied  to  any  function  is  equivalent  to  changing  x 
into  x  +  h  throughout  in  the  function. 

d_ 

Accordingly,  ehax$  (x,  y)  =  cj>(x  +  h,y),  since  y  is  indepen- 
dent of  x. 


Extension  of  Taylor's  Theorem,  1 6 1 

d 

In  like  manner,  the  operation  ekdv,  when  applied  to  any 
function,  changes  y  into  y  +  k ; 

d  d  d_ 

.-.  ek*»  .  ehTx(f>  (x,  y)  =  ekdy(j> (x  +  h,  y)  =  <j>(x  +  h,  y  +  k), 

d_       £ 

or  ehTy+hdx^  (x,  y)  =  <p  (x  +  h,  y  +  k), 

d  d 

assuming  that  the  symbols  k  —  and  h  —  are  combined  ac~ 

ay  cix 

cording  to  the  same  laws*  as  ordinary  algebraic  expressions. 

In  an  analogous  manner  we  obtain  the  symbolic  formula 

d         d        d 

eh*:+kdV+  te<j>(X)  y,  z)  =  $(x  +  h,  y  +  k,  z  +  I).  (8) 

131.  If  in  the  development  (2),  dx  be  substituted  for  h, 
and  dy  for  k,  it  becomes 

<j>{x  +  dx,y  +  dy)  =  $  +  —dx  +  -^  dy 


+  - 
1 


-2(3^+2^^+0^)+&c-  <«> 


If  the  sum  of  all  the  terms  of  the  degree  n  in  dx  and  dy 
be  denoted  by  dn$,  the  preceding  result  may  be  written  in 
the  form 

.  7  x  d6      d26         d3d> 

<b(x  +  dx,  y  +  dy)  =  d>  +  —  +  — -  + - —  +  .  .  . 

rv  r       1       1.2      1.2.3 

\n 
Since  dx,  dy,  are  infinitely  small  quantities  of  the  first 


(1  ?/  n  1£ 

*  That  this  is  the  case  appears  immediately  from  the  equations  — —  =-t-j  } 

d3u     __    d3u 
dx^dy     dydx2' 

M 


1 62  Extension  of  Taylor's  Theorem. 

order,  each  term  in  the  preceding  expansion  is  infinitely  small 
in  comparison  with  the  preceding  one. 

Hence,  since  d2<j>  is  infinitely  small  in  comparison  with 
dtp,  if  infinitely  small  quantities  of  the  second  and  higher 
orders  be  neglected  in  comparison  with  those  of  the  first,  in 
accordance  with  Art.  38,  we  get 

dip  =  <p{x  +  dx,  y  +  dy)  -  <p(x9  y)  -—^-dx-i-  -j-  dy, 

U/X  cty 

which  agrees  with  the  result  in  Art.  97. 

132.  Euler's  Theorems  of  Homogeneous  Func- 
tions.— "We  now  proceed  to  give  another  proof  of  Euler's 
Theorems  in  addition  to  those  contained  in  Arts.  102  and  123. 

If  we  substitute  gx  for  h  and  gy  for  k  in  the  expansion  (5), 
it  becomes 

,  x  f  du        du\ 

♦(.  +  «*  +  ff)..  +  ;^^) 

q2   /  „  d2u  d2u        _  d2u\     0 

+  -£—-  [x2  -77  +  zxy-r-r  +  y2  —  )+  &c, 
1 .  2  V     dx2         *  dxdy     *  dy2  J  ' 

where  u  stands  for  ip(x,  y). 

But      <p(x  +  gx,y  +  gy)  =  ip[(i  +  g)x,  (1  +  g)y} ; 

and,  if  ip  (a?,  y)  be  a  homogeneous  function  of  the  nth  degree 
in  x  and  y,  it  is  evident  that  the  result  of  substituting  (1  +  g)x 
for  x,  and  (1  +  g)  y  for  y  in  it,  is  equivalent  to  multiplying  it 
by  (1  +  g)n.     Hence,  we  have  for  homogeneous  functions, 

<p(x  +  gx,y  +  gy)  =  (1  +  g)n  ip(x,  y)  =  (i  +  g)nu, 

,         .  f  du        du\ 

or     (1  +  gyu  -  •  +  g[.-  +  y^j 

q2   (  0d2u  d2u        0d2u\      0 

^—   af—  +  zxy-^—r  +  y 3-r-r    +  &c, 

t.2\     ^  *«&#       ^   %V 


+ 

1.2 


where  w  is  a  homogeneous  function  of  the  nth  degree  in  x 
and  y. 


Enter's  Theorems.  163 

Since  the  preceding  equation  holds  for  all  values  of  g,  if 
we  expand  and  equate  like  powers  of  g,  we  obtain 


du  du 
—  +  y  — 
ax        ay 


x  —  +  y  —  =  nu9 


0d2u  d2u  d2u        ,  N 

r—  +  2W-—  +  y2—  =  n[n  -  1)  u, 
dx2  *  dxdy     ^  dy2        v  ' 

&c.  &c.  &c. 


The  foregoing  method  of  demonstration  admits  of  "being 
easily  extended  to  the  case  of  a  homogeneous  function  of  three 
or  more  variables. 

Thus,  substituting  gx  for  h,  gy  for  k,  gz  for  I,  in  formula 
(6)  Art.  129,  and  proceeding  as  before,  we  get 

du        du        du 
x—  +  y—  +  z—  =  nu, 
dx        dy        dz 

nd2u       „  d2u       „  d2u  d?u  d2u 

^77  +  y  tt  +  *ti  +  2^  ~r~r  +  22^  tt- 
dar  ay  as  aWy  asaa; 

a^         ,  x 

+  2t/z  -^r—r  =  n[n  -  i)u, 
dydz 

&c.  &c.  &c. 


These  formulae  are  due  to  Euler,  and  are  of  importance 
in  the  general  theory  of  curves  and  surfaces,  as  well  as  in 
other  applications  of  analysis. 

The  preceding  method  of  proof  is  taken  from  Lagrange's 
Mecanique  Analytique. 

M  2 


(     1 64     ) 


CHAPTEE  IX. 

MAXIMA    AND  MINIMA   OF   FUNCTIONS   OF   A   SINGLE    VARIABLE. 

133.  Definition  of  a  Maximum  or  a  Minimum. — If  any 

function  increase  continuously  as  the  variable  on  which  it  de- 
pends increases  up  to  a  certain  value,  and  diminish  for  higher 
values  of  the  variable,  then,  in  passing  from  its  increasing  to  its 
decreasing  stage,  the  function  attains  what  is  called  a  maximum 
value. 

In  like  manner,  if  the  function  decrease  as  the  variable 
increases  up  to  a  certain  value,  and  increase  for  higher  values 
of  the  variable,  the  function  passes  through  a  minimum  stage. 

Many  cases  of  maxima  and  minima  can  be  best  determined 
without  the  aid  of  the  Differential  Calculus ;  we  shall  com- 
mence with  a  few  geometrical  and  algebraic  examples  of  this 
class. 

134.  Geometrical  Example. — To  find  the  area  of  the 
greatest  triangle  which  can  be  inscribed  in  a  given  ellipse.  Sup- 
pose the  ellipse  projected  orthogonally  into  a  circle ;  then  any 
triangle  inscribed  in  the  ellipse  is  projected  into  a  triangle 
inscribed  in  the  circle,  and  the  areas  of  the  triangles  are  to 
one  another  in  the  ratio  of  the  area  of  the  ellipse  to  that  of 
the  circle  (Salmon's  Conies,  Art.  368).  Hence  the  triangle  in 
the  ellipse  is  a  maximum  when  that  in  the  circle  is  a  maxi- 
mum ;  but  in  the  latter  case  the  maximum  triangle  is  evidently 
equilateral,  and  it  is  easily  seen  that  its  area  is  to  that  of  the 
circle  as  ^27  to  471-.  Hence  the  area  of  the  greatest  triangle 
inscribed  in  the  ellipse  is 

Zab^/i 


where  a9  b  are  the  semiaxes. 

Moreover,  the  centre  of  the  ellipse  is  evidently  the  point 
of  intersection  of  the  bisectors  of  the  sides  of  the  triangle. 


Algebraic  Examples  of  Maxima  and  Minima.  165 

Examples. 

1.  Prove  that  the  area  of  the  greatest  ellipse  inscribed  in  a  given  triangle  is 
ir 

/ —  (area  of  the  triangle). 

2.  Find  the  area  of  the  least  ellipse  circumscribed  to  a  given  triangle. 

3.  Place  a  chord  of  a  given  length  in  an  ellipse,  so  that  its  distance  from  the 
centre  shall  be  a  maximum. 

The  lines  joining  its  extremities  to  the  centre  must  be  conjugate  diameters. 

4.  Show  that  the  preceding  construction  is  impossible  when  the  length  of 

the  given  chord  is  >a\/ '2  or  <b\/2  ;  where  a  and  b  are  the  semiaxes  of  the 
ellipse.  Prove  in  this  case  that  if  the  distance  of  the  chord  from  the  centre  be 
a  maximum  or  a  minimum  the  chord  is  parallel  to  an  axis  of  the  curve. 

5.  A  chord  of  an  ellipse  passes  through  a  given  point,  find  when  the  triangle 
formed  by  joining  its  extremities  to  the  centre  is  a  maximum. 

6.  Prove  that  the  area  of  the  maximum  polygon  of  n  sides,  inscribed  in  a 

given  ellipse,  is  represented  by  -  ab  sin  — . 

2  n 

135.  Algebraic  Examples  of  Maxima  and  minima. 

— Many  cases  of  maxima  and  minima  can  be  solved  by  ordi- 
nary algebra.  We  shall  confine  our  attention  to  one  simple 
class  of  examples. 

Let/(#)  represent  the  function  whose  maximum  or  mini- 
mum values  are  required,  and  suppose  u  =  /(#),  and  solve 
for  x ;  then  the  values  of  u  for  which  x  changes  from  real  to 
imaginary,  are  the  solutions  of  the  problem.  This  method  is, 
in  general,  inapplicable  when  the  equation  in  x  is  beyond  the 
second  degree.  We  shall  illustrate  the  process  by  a  few  ex- 
amples : — 

Examples. 

1.  To  divide  a  number  into  two  parts  such  that  their  product  shall  be  a 
maximum. 

Let  a  denote  the  number,  x  one  of  the  parts,  then  x  (a  —  x)  is  to  be  a  maxi- 
mum, by  hypothesis. 

Here  u  =  x(a  -  x),  or  #2  -  ax  +  u  =  o  ; 

solving  for  x  we  get 


u         \a* 


fjp. 

accordingly,  the  maximum  value  of  u  is  — ,  since  greater  values  would  make  x 

4 
imaginary. 


1 66     Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

2.  To  find  the  maximum  and  minimum  values  of  the  fraction 


x2  +  i 


I  V    (l  —  2w) (i  +  1U) 


Here  u  =  — ,  or  #2  +  i  =  - ;  .*.  a;  = h 

XA  +  I  W  2W  ~~  2W 

In  this  case  we  infer  that  the  maximum  and  minimum  values  of  u  are  -  and 

2 

;  and  the  proposed  fraction  accordingly  lies  between  the  limits  -  and 

2  2  2 

for  all  real  values  of  x. 

These  results  can  he  also  easily  established,  as  follows.   "We  have  in  all  cases 

(x  +  yf  =  (x-  yf  +  4%y. 

Accordingly,  if  x  +  y  he  given,  xy  is  greatest  when  x  -  y  =  o,  or  when  x  =  y. 
Conversely,  if  xy  be  given,  the  least  value  of  x  +  y  is  when  x  =  y. 

a2 
Hence,  denoting  xy  by  a2,  the  minimum  value  of  x  -\ —  is  za,  for  positive 

x 
values  of  x. 

Again,  it  is  evident  that  when  a  function  attains  a  maximum  value,  its  in- 
verse becomes  a  minimum ;  and  vice  versd. 

Accordingly,  the  max.  value  of  — -  is  — ,  under  the  same  condition. 

x2  +  a2      za 

3.  Find  the  greatest  value  of 


(a  +  x)  (b  +  x) 

(a  +  x)  (b  +  x)                       .   .                 ab         .  .  /— - 

Here is  to  be  a  minimum,  or \-  x  is  a  mm. ;  .*.  x  =  y  abr 


x  x 

and  the  max.  value  in  question  is 


4- 


(*/a  +V/t>)2' 

{x  +  a)  (x  +  b) 
x  +  c 

(2  +  a  —  c)  (z  +  b  —  c) 


Let  x  +  c  =  z,  and  the  fraction  becomes 

z 

In  order  that  this  should  have  a  real  min.  value,  (a  —  e)(b  —  c)  must  be  posi- 
tive ;  i.  e.  the  value  of  e  must  not  lie  between  those  of  a  and  b,  &c» 

5.  Find  the  least  value  of  a  tan  6  +  b  cot  d.  Am.  2  \/  ab. 

6.  Prove  that  the  expression ; will  always  lie  between  two  fixed 

r  x%  +  bx  +  c~ 

finite  limits  if  a2  +  c2  >  ab  and  b2  <  4  c2 ;  that  there  will  be  two  limits  between 
which  it  cannot  lie  if  a2  +  c2  >  ab  and  b2  >  4  c2  :  and  that  it  will  be  capable  of  all 
values  if  a2  +  c2  <  ab. 

136.  To  find  the  Maximum  and  Minimum  values 
of 

ax*  +  ibxy  +  cy% 

dx%  +  ib'xy  +  cy1' 


Algebraic  Examples  of  Maxima  and  Minima.  167 


x 


Let  u  denote  the  proposed  fraction,  and  substitute  z  for  -; 

If 

then  we  get 

az2  +  ibz  +  c  .  v 

"  =  aV  +  aftW  '  (I) 

or  (a  -  a'u)z2  +  2  (b  -  bfu)z  +  c-  c'u  =  o. 

Solving  for  z,  this  gives 

(a - du)z  +  b-b'u  =  ± yy{b - b'u)2  -  (a - a'u)  {c - c'u) .         (2) 

There  are  three  cases,  according  as  the  roots  of  the  equation 

(b'2  -  a'c')  u2  +  {ad  +  ca'-  2bb')u+b2-ac  =  o  (3) 

are  real  and  unequal,  real  and  equal,  or  imaginary. 

(1).  Let  the  roots  be  real  and  unequal,  and  denoted  by 
a  and  (3  (of  which  (3  is  the  greater) ;  then,  if  V2  -  a'c'  >  o,  we 
shall  have 


(a-a'u)z  +  b-b'u  =  ±  y(b'2  -  a'c')  (u-a)  (u-[3). 

Here,  so  long  as  u  is  not  greater  than  a,  z  is  real ;  but 
when  u  >  a  and  <  ]3,  z  becomes  imaginary ;  consequently,  the 
lesser*  root  (a)  is  a  maximum  value  of  u.  In  like  manner,  it 
can  be  easily  seen  that  the  greater  root  (j3)  is  a  minimum. 

Accordingly,  when  the  roots  of  the  denominator,  a'x2  +  2b' x 
+  cf  =  o,  are  real  and  unequal,  the  fraction  admits  of  all  pos- 
sible, positive,  or  negative  values,  with  the  exception  of  those 
which  lie  between  a  and  ]3. 

If  either  a'  -  o,  or  c'  =  o,  the  radical  becomes 


b'  y(u  -a)  (u-  j3), 

and,  as  before,  the  greater  root  is  a  minimum,  and  the  lesser 
a  maximum,  value  of  u. 

*  In  general,  in  seeking  the  maximum  or  minimum  values  of  y  from  the 
equation,  y  =  <j>(%),  if  for  all  values  of  y  between  the  limits  a  and  j8,  the  corre- 
sponding values  of  x  are  imaginary,  while  x  is  real  when  y  =  a,  or  y  =  #  ;  then 
it  is  evident  that  the  lesser  of  the  quantities,  o,  j8,  is  a  maximum,  and  the  greater 
a  minimum,  value  of  y.  This  result  also  admits  of  a  simple  geometrical  proof, 
by  considering  the  curve  whose  equation  is  y  =  <p(x). 


1 68     Maxima  and  Minima  of  Functions  of  a  Bingle  Variable. 

(2.)  When  a  =  j3,  the  expression  under  the  radical  sign  is 
positive  for  all  values  of  u,  and  consequently  u  does  not  admit 
of  either  a  maximum  or  a  minimum  value. 

(3.)  "When  the  roots  a  and  j3  are  imaginary,  the  expres- 
sion under  the  radical  sign  is  necessarily  positive,  and  u  in 
this  case  also  does  not  admit  of  either  a  maximum  or  a  mini- 
mum value. 

Hence,  in  the  two  latter  cases,  the  fraction  admits  of  all 
possible  values  between  +  co  and  -  co  . 

In  the  preceding,  the  roots  of  the  denominator  are  sup- 
posed real ;  if  they  be  imaginary,  i.e.  if  6'2  -  dd  <  o,  we  have 


(a-au)z  +  b-b'u  =  ±y(a'c'-b'2)  (u-a)  (j5-u). 

It  is  easily  seen  that  z  is  imaginary  for  all  values  of  u 
except  those  lying  between  a  and  j3.  Accordingly,  the  greater 
root  is  a  maximum,  and  the  lesser  a  minimum,  value  of  u. 

Hence,  in  this  case,  the  fraction  represented  by  u  lies  be- 
tween the  limits  a  and  ]3  for  all  real  values  of  x  and  y. 

137.  Ctuadratic  for  determining  z. — Again,  the  value 
of  z,  corresponding  to  a  maximum  or  a  minimum  value  of  u, 
must  satisfy  the  equation 

(a  -  du)z  +  b  -  b'u  =  o. 

Substituting  for  u  in  (1)  its  value  derived  from  this  latter 
equation,  we  obtain  the  following  quadratic  in  z  : 

(ab'  -  bd)  z*  +  z  (ad  -  cd)  +  bd  -cb'  =  o.  (4) 

This  equation  determines  the  values  of  z  which  correspond 
to  the  maximum  and  minimum  values  of  u.  It  can  be  easily 
seen  that  if  the  roots  of  equation  (3)  are  real  so  also  are  those 
of  (4) ;  and  vice  versa. 

The  student  will  observe  in  the  preceding  investigation 
that  when  u  attains  a  maximum  or  a  minimum  value,  the 
corresponding  equation  in  z,  obtained  from  (2),  has  equal 
roots.  This  is,  as  will  be  seen  more  fully  in  the  next  Article, 
the  essential  criterion  of  a  maximum  or  a  minimum  value,  in 
general. 


Condition  for  a  Maximum  or  Minimum.  169 

Find  the  maximum  or  minimum  values  of  u  in  the  follow- 
ing cases : — 

Examples. 

X2  +  7.X  +  1 1  .  5 

1.  u  = .  Ans.  u  =  2,  a  max.,  w  =  —  a  mm. 

«2  +  42;  +  10  o 

#2  —  X  +   l  7  —  IX 

2.  U  =  — =    I  + 


#a  +  #  —  I  X*  +  X—  I 

T    —  5*  #      -4-  3/  ~~    I 

is  a  max.  or  a  min.  according  as is  a  min.  or  a  max.,  i.  e. 


xi  -\-  x  —  1  1  —  # 

1 

as #  is  a  maximum  or  a  minimum. 

1  —  x 

o\  #=  o,  or  x  =  2  ;  the  former  gives  a  maximum,  the  latter  a  minimum  solution. 

We  now  proceed  to  a  general  investigation  of  the  condi- 
tions for  a  maximum  and  minimum,  by  aid  of  the  principles 
of  the  Differential  Calculus. 

138.   Condition  for  a  Maximum  or  Minimum. — If 

the  increment  of  a  variable,  x,  be  positive,  then  the  corre- 
sponding increment  of  any  function,  f(x),  has  the  same  sign 
as  that  of  f\x),  by  Art.  6  ;  hence,  as  x  increases,  ,/(#)  increases 
or  diminishes  according  as /'(a?)  is  positive  or  negative. 

Consequently,  tvhen  f{x)  changes  from  an  increasing  to  a 
decreasing  state,  or  vice  versa,  its  derived  function  ff(x)  must 
change  its  sign.  Let  a  be  a  value  of  x  corresponding  to  a 
maximum  or  a  minimum  value  of  f{x) ;  then,  in  the  case  of 
a  maximum  we  must  have  for  small  values  of  h, 

f(a)  >f(a  +  h),  and/(a)  >f{a-h) ; 

and,  for  a  minimum, 

f{a)  <f(a  +  h),  and/(#)  <f(a-h). 

Accordingly,  in  either  case  the  expressions 

f(a  +  h)-f(a),  tmdf(a-h)-f(a), 

have  both  the  same  sign. 


170    Maxima  and  Minima  of  Functions  of  a  Single  Variable. 
Again,  by  formulae*  (29),  Art.  75,  we  have 
f(a  +  h)  -f(a)  =  hf(a)  +  ~f(*  +  Oh), 

f(a  -  h)  -f(a)  =  -  hf'(a)  +  -^J"(a  -  61h). 

Now,  when  h  is  very  small,  and  f'ia)  finite,  the  second 
term  in  the  right-hand  side  in  each  of  these  equations  is  very 
small  in  comparison  with  the  first,  and  hence  f{a  +  h)  -f(a) 
and  f(a-  h)  -  f(a)  cannot  have  the  same  sign  unless 
f(a)  =  o. 

Hence,  the  values  of  %  which  render  f(x)  a  maximum  or  a 
minimum  are  in  general  roots  of  the  derived  equation  fix)  =  o. 

This  result  can  also  be  arrived  at  from  geometrical 
considerations ;  for,  let  y  =  fix)  be  the  equation  of  a  curve, 
then,  at  a  point  from  which  the  ordinate  y  attains  a  maximum 
or  a  minimum  value,  the  tangent  to  the  curve  is  evidently 
parallel  to  the  axis  of  x ;  and,  consequently  fix)  =  o,  by 
Art.  10. 

Moreover,  if  x  be  eliminated  between  the  equations 
fx)  =  u  and/'^)  =  o,  the  roots  of  the  resulting  equation  in 
u  are,  in  general,  the  maximum  and  minimum  values  of  f[x). 

This  is  the  extension  of  the  principle  arrived  at  in 
Art.  134. 

•  Again,  since  f(a)  =  o,  we  have 


f(a  +  h)-Aa)  =  ~f'(a  +  eh), 
f(a-hy-f(a)=—f'(a-M) 


(5) 


*  In  the  investigation  of  maxima  and  minima  given  above,  Lagrange's  form 
of  Taylor's  Theorem  has  been  employed.  For  students  who  are  unacquainted 
with  this  form  of  the  Theorem,  it  may  be  observed  that  the  conditions  for  a 
maximum  or  minimum  can  be  readily  established  from  the  form  of  Taylor's 
Series  given  in  Art.  54,  viz., 

h2  h3 

f(a  +  h)  -/(«)  =  hf(a)  +  —f\a)  + /"»  +  &c. ; 

1.2  1.2.3 

for  when  h  is  very  small  and  the  coefficients/^),/"  (a),  &c.  finite,  it  is  evident 
that  the  sign  of  the  series  at  the  right-hand  side  depends  on  that  of  its  first 
term,  and  hence  all  the  results  arrived  at  in  the  above  and  the  subsequent 
Articles  can  be  readily  established. 


Condition  for  a  Maximum  or  a  Minimum. 


171 


But  the  expressions  at  the  left-hand  side  in  these  equations 
are  both  positive  for  small  values  of  h  when/"(#)  is  positive ; 
and  negative,  when  f\a)  is  negative ;  therefore  f(a)  is  a 
maximum  or  a  minimum  according  as  f\a)  is  negative  or 
positive. 

If,  however,  f'(a)  vanish  along  with  f(a),  we  have,  by 
Art.  75, 

f(a  +  h)  -/(«)  =_!_/'»  +  -_£_/*(*  +  Oh), 

f{a  -  h)  -fid)  =  -if'W  +        *;      /*(<*  -  W). 

Hence  it  follows  that  in  this  case,  /(a)  «'s  neither  a 
maximum  nor  a  minimum  unless  f'"(a)  also  vanish;  but  if 
f"{a)  =  o,  then  f(a)  is  a  maximum  when  fiY(a)  is  negative, 
and  a  minimum  when/iv(a)  is  positive. 

In  general,  let /(")(#)  be  the  first  derived  function  that 
does  not  vanish  ;  then,  if  n  be  odd,  f(a)  is  neither  a  maximum 
nor  a  minimum ;  if  n  be  even,  f(a)  is  a  maximum  or  a  mini- 
mum according  as/(w+1)(#)  is  negative  or  positive. 

The  student  who  is  acquainted  with  the  elements  of  the 
theory  of  plane  curves  will  find  no  difficulty  in  giving  the 
geometrical  interpretation  of  the  results  arrived  at  in  this 
and  the  subsequent  Articles. 

Examples. 

i.     u=  a  sinx  +  b  coax. 

Here  the  maximum  and  minimum  values  are  given  by  the  equation 

du  a 

—  =  a  cos  x  —  o  sin  x  =  o,  or  tan  x  =  7. 
dx  0 

Hence,  the  max.  value  of  u  is  y  a2  +  b2,  and  the  min.  is  —  v  a2  +  b2.    This  is 
also  evident  independently,  since  u  may  be  written  in  the  form 

*f  a2  -Y  b2  sin  {x  +  a), 
b 
where  tan  a  =  — . 
a 

2.     u  =  x  —  sin  it. 

du  d2u       ,  d3u 

In  this  case   —  =  1  —  cos  x,     — -  =  sin  x,     -7-3  =  cos  #. 

dx  dx2  dx6 


172     Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

Accordingly,  if    -=o,ve  have  —  =  o,  and  —  =  1. 

Consequently,  the  function  x  —  sin  x  does  not  admit  of  either  a  maximum  or  a 
minimum  value. 

This  result  can  also  be  easily  seen  from  geometrical  considerations. 

3.     u  =  a  cos  x  +  b  cos  2x,  a  and  £  being  both  positive. 

Here  —  =  -  a  smx  -  ib  sin  ix, 

ax 

d2u 

— -  =  —  a  cos  x  —  40  cos  2#. 

The  maximum  and  minimum  values  are  given  by  the  equation  a  sin#  +  2b 

sin  ix  =  o : 

—  & 
.'.  -we  have,  (1),  sin  #  =  o ;     or  (2),  cos  x  =  — . 

40 

The  simplest  solution  of  (1)  is  x  =  0,  in  which  case 

.      clhc 
u  —  a  +  0,     —  =  —  a  —  40  ; 
dxl 

consequently  this  gives  a  maximum  solution. 

cftu 
Again,  let  x  =  ir,  and  we  have  u  —  b  —  a,  —r  =  a  —  4b  ;  consequently  this 

gives  a  maximum  or  a  minimum  solution,  according  as  a  is  <  or  >  40. 

TO  I  ^2^ 

If  a  =  40.  we  get  when  x  =  ir,  — -  =  o. 

On  proceeding  to  the  next  differentiation  we  have 

— -g  =  « (sin  2;  +  2  sin  22),  =  o  when  #  =  tt. 

d% 

Again,    —  =  a  (cos  x  +  4  cos  2#)  =  3^.       Consequently  the  solution  is  a 
$># 

minimum  m  this  case. 

Again,  the  solution  (2)  is  impossible  unless  a  be  less  than  40.     In  this  case, 

d2u 
i.  e.  when  a  <  40,  we  easily  find  — ■  positive,  and  accordingly  this  gives  a  min. 

ctx 

value  of  u,  viz.  -  —  -J. 
80 

4.  Find  the  value  of  x  for  which  sec  x  —  x  is  a  maximum  or  a  minimum. 

Ans.  sins  = . 


Application  to  Rational  Algebraic  Expressions.        173 

139.  Application  to  Rational  Algebraic  Expres- 
sions.— Suppose  fix)  a  rational  function  containing  no 
fractional  power  of  at,  and  let  the  real  roots  of  fix)  =  o, 
arranged  in  order  of  magnitude,  be  a,  j3,  y,  &c. ;  no  two  of 
which  are  supposed  equal. 

Then       fix)  =  (x  -  a)  [x  -  j3)  (x  -  y)  .  .  . 
and  /"(«)=  (<*-0)  (a-y)  .  .  . 

But  by  hypothesis,  a  -  j3,  a-y,  &c.  are  all  positive ;  hence 
/"(a)  is  also  positive,  and  consequently  a  corresponds  to  a 
minimum  value  oif(x). 

Again,  /"(/3)  =  (fi  -  «)  (0  -  y) 

here  j3  -  a  is  negative,  and  the  remaining  factors  are  positive ; 
hence  f'((3)  is  negative,  and/(|3)  a  maximum. 
Similarly,/ (7)  is  a  minimum,  &c. 

140.  Maxima  and  Minima  Values  occur  alter- 
nately.— We  have  seen  that  this  principle  holds  in  the  case 
just  considered. 

A  general  proof  can  easily  be  given  as  follows  : — Suppose 
fix)  a  maximum  when  x  =  a,  and  also  when  x  =  b,  where  b  is 
the  greater  ;  then  when  x  =  a  +  h,  the  function  is  decreasing, 
and  when  x  =  b  -  h,  it  is  increasing  (where  h  is  a  small  incre- 
ment) ;  but  in  passing  from  a  decreasing  to  an  increasing 
state  it  must  pass  through  a  minimum  value ;  hence  between 
two  maxima  one  minimum  at  least  must  exist. 

In  like  manner  it  can  be  shown  that  between  two  minima 
one  maximum  must  exist. 

141.  Case  of  Equal  Roots. — Again,  if  the  equation 
fix)  =  o  has  two  roots  each  equal  to  a,  it  must  be  of  the  form 

fix)  =  ix  -  a)2  ip  (x). 

In  this  case  /"(a)  =  o,f"(a)  =  2\p(a),  and  accordingly, 
from  Art.  138,  a  corresponds  to  neither  a  maximum  nor  a 
minimum  value  of  the  function /(#). 

In  general,,  if  fix)  have  n  roots  equal  to  a,  then 

f(x)  =  {x-a)n4,(x). 

Here,  when  n  is  even,  /(a)  is  neither  a  maximum  nor  a 
minimum  solution :  and  when  n  is  odd,  f(a)  is  a  maximum  or 
a  minimum  according  as  \p(a)  is  negative  or  positive. 


174     Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

14.2.  Case  where  fix)  =  00.  The  investigation  in 
Art.  138  shows  that  a  function  in  general  changes  its  sign  in 
passing  through  zero. 

In  like  manner  it  can  be  shown  that  a  function  changes 
its  sign,  in  general,  in  passing  through  an  infinite  value  ;  i.e.  if 
0(a)  =  co,  0  (a  -  h)  and  <f>(a+h)  have  in  general  opposite  signs, 
for  small  values  of  h. 

For,  if  u  and  -  represent  any  function  and  its  reciprocal, 

they  have  necessarily  the  same  sign ;  because  if  u  be  positive, 

-  is  positive,  and  if  negative,  negative. 

u 

Suppose  Ui,  u2,  uz,  three  successive  values  of  u,  and 
— ,    — ,    — ,  the  corresponding  reciprocals. 

Ui       U2       U3  r  o  r 

Then,  if  u2  =  o,  by  Art.  138,  ux  and  u6  have  in  general 
opposite  signs. 

Hence,  if  —  =  co  ,  —  aud—  have  also  opposite  signs;  and 

U-i  U\  u$ 

we  infer  that  the  values  of  x  which  satisfy  the  equation  fix) 
=  00  may  furnish  maxima  and  minima  values  of  fix). 

143.  "We  now  return  to  the  equation 

f(x)  =  (x-a)nxp(x), 

in  which  n  is  supposed  to  have  any  real  value,  positive,  nega- 
tive, integral,  or  fractional. 

In  this  case,  when  x  =  a,f  (x)  is  zero  or  infinity  according 
as  n  is  positive  or  negative. 

To  determine  whether  the  corresponding  value  of  fix)  is 
a  real  maximum  or  minimum,  we  shall  investigate  whether 
fix)  changes  its  sign  or  not  as  x  passes  through  a. 

When  x  =  a  +  h,    /{a  +  h)  =  hn  \p  (a  +  h), 

x  =  a-h,    f(a-h)  =  (-h)nip(a-h): 


J5 


now,  when  h  is  infinitely  small,  \p(a  +h)  and  \p  (a  -  h)  become 
each  ultimately  equal  to  \fj  (a) :  and  therefore  f\a  +  h)  and 
f\a  -  h)  have  the  same  or  opposite  signs  according  as  ( -  1)" 
is  positive  or  negative. 


Examples.  175 

(1).  If  n  be  an  even  integer,  positive  or  negative,  ./"(a?)  does 
not  change  sign  in  passing  through  a,  and  accordingly  a  cor- 
responds to  neither  a  maximum  nor  a  minimum  solution. 

(2).  If  n  be  an  odd  integer,  positive  or  negative,  f(a  +  h) 
and/"(«  -  h)  have  opposite  signs,  and  a  corresponds  to  a  real 
maximum  or  minimum. 


ir 
+  — 

2r 


2r  p 

(3).  If  n  be  a  fraction  of  the  form  ±  — ,  then  (  -  1) 


r 

+  - 


=  1  p  =  i ,  and  a  corresponds  to  neither  a  maximum  nor  a 
minimum. 

2r  +  i  £ 

(4).  If  n  be  of  the  form  ± -,  then  (  -  1)        =  (- 1)    ; 

jj 

this  is  imaginary  Up  be  even,  but  has  a  real  value  ( -  1)  when 
p  is  odd.  In  the  former  case,  fr(a  —  h)  becomes  imaginary  ;  in 
the  latter,  f'(a  +  h)  aji&f'(a-h)  have  opposite  signs,  and  f(a) 
is  a  real  maximum  or  minimum. 

Thus  in  all  cases  of  real  maximum  and  minimum  values 
the  index  n  must  be  the  quotient  of  two  odd  numbers. 


Examples. 

1.  f(x)  =  ax2  +  2bx  +  c. 

b 
Here  f(x)  -  i{ax  +  b)  =  o ;         hence  cs  = , 

f'\x)  =  2a. 

ft/*  Jy»  w 

And is  a  maximum  or  a  minimum  value  of  ax2  +  2bx  +  c,  according 

a 
as  a  is  negative  or  positive. 

2.  f{x)  =  22?  -  i$x2  +  36a;  +  10. 

Here  /'  (*)  =  6(*2  -  5^  +  6)  =  6(*  -  2)  (s  -  3). 

(r.)  Let  x  =  2  ;  then /"(#)  is  negative  ;  hence /(2)  or  38  is  a  maximum. 
(2.)  Let  x  =  3  ;  then/  "(#)  is  positive;  hence  /  (3)  or  37  is  a  minimum. 


176    Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

It  is  evident  that  neither  of  these  values  is  an  absolute  maximum  or  mini- 
mum ;  for  when  x  =  00  ,  f(x)  =  00  ,  and  when  x  -  -  00  ,  f(x)  =  -  00  ;  accord- 
ingly, the  proposed  function  admits  of  all  possible  values,  positive  or  negative. 

Again,  neither  +  00  nor  —  00  is  a  proper  maximum  or  minimum  value,  because 
for  large  values  of  #,  f{x)  constantly  increases  in  one  case,  and  constantly  dimi- 
nishes in  the  other. 

It  is  easily  seen  that  as  x  increases  from  -  00  to  +  2,  f(x)  increases  from  -  00 
to  38  ;  as  #  increases  from  2  to  3,/0)  diminishes  from  38  to  37  ;  and  as  x  in- 
creases from  3  to  00,  f(x)  increases  from  37  to  00.  When  considered  geome- 
trically, the  preceding  investigation  shows  that  in  the  curve  represented  by  the 
equation 

y  =  2X3  —  I$X2  +  $6x  +  IO, 

the  tangent  is  parallel  to  the  axis  of  x  at  the  points  x  =  2,  y  =  38  ;  and  x  =  3, 
y  —  37  ;  and  that  the  ordinate  is  a  maximum  in  the  former,  and  a  minimum  in 
the  latter  case,  &c. 

3.  f{x)  =  a  +  b  (x  —  c)i.  Am.  x  =  c.     Neither  a  max.  nor  a  min. 

4.  f{x)  =  b  +  e{x  -  «)§  +  d{x  -  a)i. 
Substitute  a  +  h  f or  x,  and  the  equation  becomes 

f(a  +  h)  =  b  +  c0  +  dJ&\ 

also  /(«  -  h)  —  b  +  c0  +  dh? ; 

but  when  h  is  very  small  h*  is  very  small  in  comparison  with  hi,  and  accordingly  b 
is  a  minimum  or  a  maximum  value  of /(#)  according  as  c  is  positive  or  negative. 

5.  /(#)  =  5#6  +  I2%5  -  ISxi  —  4oa;3  +  ISX%  +  6o#  +  J7' 

Ans.  x  =  ±  1  gives  neither  a  max.  nor  a  min. ;  x  —  —  2  gives  a  min. 

6.  - — -  .    Let  x  —  10  =  z,  and  the  fraction  becomes 

x  —  10 

z  z 

36 

The  maximum  and  minimum  values  are  given  by  the  equation  1 =  o; 

.*.  z  =  +  6,  and  hence  x  =  16  or  4;  the  former  gives  a  minimum,  the  latter 
a  maximum  value  of  the  fraction. 

*,  n    O  -  03 

Hence  f(m)  =  ^=^  (0  +  5). 

If  #  =  1,  /(#)  is  neither  a  maximum  nor  a  minimum ;  if  x  =  —  5,  /(#)  is  a 
maximum. 


7  vr.       /.  «a?  +  ^bccv  +  cy2  177 

ife.  and  Mm,  of  —o ~ j-. .  ' ' 

ax*  +  2b  xy  +  cy2 

(x  +  i)2 

Again,  the  reciprocal  function  7 rz  is  evidently  a  max.  when  %  —  -  1 ; 

{%  -  i)6 

for  if  we  substitute  for  x,  —  1  +  h,  and  -  1  —  h,  successively,  the  resulting 

values  are  both  negative  ;  and  consequently  the  proposed  function  is  a  minimum 

in  this  case. 

This  furnishes  an  example  of  a  solution  corresponding  to  f'{%)  =0°.    See 

Art.  142. 

144.  We  shall  now  return  to  the  fraction 

ax2  +  2bxy  +  cy* 
a'x2  +  2b' xy  +  c'y2' 

the  maximum  and  minimum  values  of  which  have  been  already 
considered  in  Art.  136. 

Write  as  before  the  equation  in  the  form 

z2{a  -  a'u)  +  2%(b  -  b'u)  +  (c  -  c'u)  =  o, 

x 
where  2  =  -. 

y 

dt/ijfi 

Differentiate  with  respect  to  2,  and,  as  —  =  o  for  a  maxi- 

az 

mum  or  a  minimum,  we  have 

2  (a  -  a'u)  +  (b  -  b'u)  =  o. 

Multiply  this  latter  equation  by  2,  and  subtract  from  the 
former,  when  we  get 

z(b  -  b'u)  +  (c  -  c'u)  -  o. 
Hence,  eliminating  2  between  these  equations,  we  obtain 
{a  -  a'u)  (c  -  c'u)  =  (b  -  b'u)29 
or        u2(a'c'  -  b'2)  -  u(ac  +  ca'  -  20b')  +  (ac  -  b2)  =  o ;     (3) 

the  same  equation  (3)  as  before. 
The  quadratic  for  2, 

z2(ab'  -  bd)  +  z(ac'  -  ca')  +  be'  -  cb'  =  o,  (4) 

is  obtained  by  eliminating  u  from  the  two  preceding  linear 
equations. 

N 


178     Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

This  equation  can  also  be  written  in  a  determinant  form, 
as  follows : — 


I 

-z 

z2 

a 

b 

c 

a' 

V 

c' 

=  o. 


It  may  be  observed  that  the  coefficients  in  (3)  are  in- 
variants  of  the  quadratic  expressions  in  the  numerator  and 
denominator  of  the  proposed  fraction,  as  is  evident  from  the 
principle  that  its  maximum  and  minimum  values  cannot  be 
altered  by  linear  transformations. 

This  result  can  also  be  proved  as  follows : — 


T  aX2+2bXY+cY2 

.Let  u  = 


a'X2  +  2V  XY  +  c'Y2i 

where  X,  Y  denote  any  functions  of  x  and  y ;  then  in  seeking 
the  maximum  and  minimum  values  of  u  we  may  substitute 

%  for  — ,  when  it  becomes 

az2  +  ibz  +  c 

az2  +  2bz  +  c 

and  we  obviously  get  the  same  maximum  and  minimum  values 
for  u,  whether  we  regard  it  as  determined  from  the -original 
fraction  or  from  the  equivalent  fraction  in  z. 

Again,  let  X,  Y  be  linear  functions  of  x  and  y,  i.  e. 

X  =  Ix  +  my,     Y=  I'x  +  ni'y, 

then  u  becomes  of  the  form 

Ax2  +  2JBxy  +  Cy2 
A'x2  +  2&xy  +  C'y2' 

where  A,  B,  C,  A',  B',  (7,  denote  the  coefficients  in  the  trans- 
formed expressions  ;  hence,  since  the  quadratics  which  deter- 
mine the  maximum  and  minimum  values  of  u  must  have  the 
same  roots  in  both  cases,  we  have 

AC  -  B2  =  X(ac  -  b2),  AC  +  OA!  -  2BF  =  \(ac'  +  ca'  -  2bb'), 

A'C  -B>2  =  \{a'c'  -  V2).  Q.E.D. 


Application  to  Surfaces.  179 

It  can  be  seen  without  difficulty  that 

A  =  {M  -  ml')2. 

We  shall  illustrate  the  use  of  the  equations  (3)  and  (4)  by 
applying  them  to  the  following  question,  which  occurs  in  the 
determination  of  the  principal  radii  of  curvature  at  any  point 
on  a  curved  surface. 

145.  To  find  the  Maxima  and  Minima  Values  of 

r  cos2a  +  2s  cos  a  cos/3  +  ^  cos2j3, 

where  cos  a  and  cos  ]3  are  connected  by  the  equation 

(1  +  p2)  COS2a  +  2pq  COS  a  COS  |3  +  (1  +  q2)  eos2/3  =  I, 

and  p,  q,  r,  s,  t  are  independent  of  a  and  j3. 

Denoting  the  proposed  expression  by  u,  and  substituting 
„     cos  a  . 

zior^TR>  we  get 
cosp 

rz2  +  2sz  +  t 
u  = 


(1  +  p2)z2  +  zpqz  +  (1  +  q2)' 

The  maximum  and  minimum  values  of  this  fraction,  by 
the  preceding  Article,  are  given  by  the  quadratic 

u2[i  +p2  +  q2)-u{(i  +q2)r  -  2pqs  +  (1  +p2)t}  +rt  -s2  =  o;    (6) 

while  the  corresponding  values  of  z  or  ^  are  given  by 

s2{(i  +p2)s  - pqr)  +  z[(i  +  p2)t  -  (1  +  q2)r} 

+  {pft  -  (1  +  ^2)s}  =  o.*        (7) 

The  student  will  observe  that  the  roots  of  the  denominator 
in  the  proposed  fraction  are  imaginary,  and,  consequently,  the 
values  of  the  fraction  lie  between  the  roots  of  the  quadratic 
(6),  in  accordance  with  Art.  136. 


*  Lacroix,  Dif.  Cal.,  pp.  575,  576. 

TsT  2 


180    Maxima  and  Minima  of  Functions  of  a  Single  Variable, 

146.  To  find  the  Maximum  and  Minimum  Radius 
Vector  of  the  Ellipse 

ax2  +  zbxy  +  cy2  =  1. 
(1).  Suppose  the  axes  rectangular ;  then 

r2  -  x2  +  y2  is  to  be  a  maximum  or  a  minimum. 

x 
Let  -  =  z,  and  we  get 

y 


2  _     s2  +  1 


az2  +  2bz  +  c 


Hence  the  quadratic  which  determines  the  maximum  and 
minimum  distances  from  the  centre  is 

r*  (ae  -  b2)  -  r2  (a  +  c)  +  1  =  o. 

The  other  quadratic,  viz. 

bx2  -  (a  -  c)xy  -  by2  =  o, 

gives  the  directions  of  the  axes  of  the  curve. 

(2.)  If  the  axes  of  co-ordinates  be  inclined  at  an  angle  w9 
then 

r2  =  x2  +  y2  +  2xy  cos  &> 
z2  +  2%  cos  w  +  1 

—  .   • 

az2  +  2&3  +  c     ' 

and  the  quadratic  becomes  in  this  case 

r* (ac  -  b2)  -  r2  (a  +  c  -  2b  cos  10)  +  sin2w  =  o, 

the  coefficients  in  which  are  the  invariants  of  the  quadratic 
expressions  forming  the  numerator  and  denominator  in  the 
expression  for  r2. 

The  equation  which  determines  the  directions  of  the  axes 
1  the  conic  can  also  be  easily  written  down  in  this  case. 


Maximum  and  Minimum  Section  of  a  Eight  Cone.      1 8 1 


147.  To  investigate  the  Maximum  and  Minimum 
Values  of 

ax3  +  ^bx2y  +  $cxy2  +  dy3 


dx2  +  $brx%y  +  3c' xy2  +  d'y 


'a/3* 


x 


Substituting  %  for  -,  and  denoting  the  fraction  by  u,  we  have 

if 

az3  +  $bz2  +  $cz  +  d 
~  dz3  +  $b'z2  +  $c'z  +  dr 

Proceeding,  as  in  Art.  144,  we  find  that  the  values  of  u  and  2 
are  given  by  aid  of  the  two  quadratics 

az2  +  2bz  +  c  =  (dz2  +  2b 'z  +  c')u, 

bz2  +  2cz  +  d=  (bfz2  +  2cz  +  d')u. 

Eliminating  u  between  these  equations,  we  get  the  following 
biquadratic  in  z  : — 

z^(abr  -  bd)  +  2zz(acf  -  cd)  +  z2{adf  -  dd  +  3 (be'  -  cb')} 

+  2z(bd'  -  db')  +  (edf  -  c'd)  =  o.  (8) 

Eliminating  z  between  the  same  equations,  we  obtain  a 
biquadratic  in  u,  whose  roots  are  the  maxima  and  minima 
values  of  the  proposed  fraction.  Again,  as  in  Art.  144,  it 
can  easily  be  shown  that  the  coefficients  in  the  equation  in  u 
are  invariants  of  the  cubics  in  the  numerator  and  denominator 
of  the  fraction. 

148.  To  eut  the  Maximum  and  Minimum  Ellipse 
from  a  Right  Cone  which  stands  on  a  given  circular 
base. — Let  AD  represent  the  axis  of 
the  cone,  and  suppose  BP  to  be  the 
axis  major  of  the  required  section;  0 
its  centre ;  a,  b,  its  semi-axes.  Through 
0  and  P  draw  LM  and  PR  parallel  to 
BO.  Then  BP  =  20,  b2  =  LO  .  OM 
(Euclid,  Book  111.,  Pr.  35)  ;  but  LO 

=™   0M=—;  .-.  b2=-.BC.  PP. 

2  2  4 

Hence  BP2  .  PR  is  to  be  a  maximum 

or  a  minimum.  s*  ^ 


1 82     Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

Let  L  BAP  =  a,  PBO  =  9,  BG  =  c. 
Then  BP  =  BCsinBCF        CC0Sa 


sin BPC     cos  (0 -a)' 

sin  PJ9i2  _  c  cos  (0  +  a) 
-  -^ginp^  =  Cos  (0  -  «) ; 

COS  (0  +  a)  . 

•"•  w  =  — ttt\ (  is  a  maximum  or  a  minimum. 

cos3  (0  -  a) 

TT  dw      sin 20-  2  sin 2a  .      n 

-tLence    -^  = — — —  =  o ;     .\  sin  20  =  2  sm 2a. 

dO         cos4  (0  -  a) 

The  solution  becomes  impossible  when  2  sin  ia  >  1  ;  i.e.  if 
the  vertical  angle  of  the  cone  be  >  300. 

The  problem  admits  of  two  solutions  when  a  is  less  than 
1 5°.     For,  if  0!  be  the  least  value  of  9  derived  from  the 


7T 


equation  sin  20  ^=  2  sin  2  a  ;  then  the  value 0i  evidently 

gives  a  second  solution. 

Again,  by  differentiation,  we  get 

d2u  2  COS 20 

1^5  =  — iTTi \  (when  sin 20  =  2  sm 2a). 

cftr     cos4  (0  -  a)  x  y 

This  is  positive  or  negative  according  as  cos  2  0  is  positive  or 
negative.  Hence  the  greater  value  of  0  corresponds  to  a 
maximum  section,  and  the  lesser  to  a  minimum. 

^  In  the  limiting  case,  when  a  =  150,  the  two  solutions 
coincide.  However,  it  is  easily  shown  that  the  corresponding 
section  gives  neither  a  maximum  nor  a  minimum  solution  of 
the  problem.     For,  we  have  in  this  case  0  =  45 ° ;  which  value 

d  if 
gives  -r^  =  o.     On  proceeding  to  the  next  differentiation,  we 

find,  when  0  =  450, 

d3u  -  4  64 

W  =  cos4(45°-a)  =  "  "9" 

Hence  the  solution  is  neither  a  maximum  nor  a  minimum. 
When  a  >  1 50,  both  solutions  are  impossible. 


Geometrical  Examples.  183 

149.  The  principle,  that  when  a  function  is  a  maximum 
or  a  minimum  its  reciprocal  is  at  the  same  time  a  minimum 
or  a  maximum,  is  of  frequent  use  in  finding  such  solutions. 

There  are  other  considerations  by  which  the  determina- 
tion of  maxima  and  minima  values  is  often  facilitated. 

Thus,  whenever  u  is  a  maximum  or  a  minimum,  so  also 

is  log  (u),  unless  u  vanishes  along  with  — . 

Again,  any  constant  may  be  added  or  subtracted,  i.e.  if 
fix)  be  a  maximum,  so  also  is/(#)  ±  c. 

Also,  if  any  function,  u,  be  a  maximum,  so  will  be  any 
positive  power  of  u,  in  general. 

150.  Again,  if  z  =  f(u),  then  dz  =  f'(u)du,  and  conse- 
quently s  is  a  maximum  or  a  minimum;  either  (1)  when 
du  =  o,  i.e.  when  u  is  a  maximum  or  a  minimum  ;  or  (2)  when 
f(u)  =  o. 

In  many  questions  the  values  of  u  are  restricted,  by  the 
conditions  of  the  problem,*  to  lie  between  given  limits; 
accordingly,  in  such  cases,  any  root  of  fiu)  =  o  does  not 
furnish  a  real  maximum  or  minimum  solution  unless  it  lies 
between  the  given  limiting  values  of  u. 

We  shall  illustrate  this  by  one  or  two  geometrical 
examples. 

(1).  In  an  ellipse,  to  find  when  the  rectangle  under  a  pair  of 
conjugate  diameters  is  a  maximum  or  a  minimum.  Let  r  be  any 
semi-diameter  of  the  ellipse,  then  the  square  of  the  conjugate 
semi-diameter  is  represented  by  a2  +  b2  -  r2,  and  we  have 

u  =  r2  (a2  +  b2  -  r2)  a  maximum  or  a  minimum. 

Here  —  =  2U2  +  b2  -  ir2) r. 

dr 

Accordingly  the  maximum  and  minimum  values  are, 
(1)  those  for  which  r  is  a  maximum  or  a  minimum  ;  i.e.  r  =  a, 
or  r  =  b ;  and,  (2)  those  given  by  the  equation 

r  (a2  +  b2  -  2r2)  =  o  ; 


*  See  Cambridge  Mathematical  Journal,  vol.  iii.  p.  237. 


1 84    Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

4 


la2  +  b2 
or  r  =  o,  and  r 


The  solution  r  =  o  is  inadmissible,  since  r  must  lie  between 
the  limits  a  and  b  :  the  other  solution  corresponds  to  the 
equiconjugate  diameters.  It  is  easily  seen  that  the  solution 
in  (2)  is  the  maximum,  and  that  in  (1)  the  minimum  value 
of  the  rectangle  in  question. 

151.  As  another  example,  we  shall  consider  the  following 
problem* : — 

Given  in  a  plane  triangle  two  sides  [a,  b)  to  find  the 
maximum  and  minimum  values  of 

1  A 

-  .  cos  — , 

C  2 

where  A  and  c  have  the  usual  significations. 

Squaring  the  expression  in  question,  and  substituting  x 
for  c,  we  easily  find  for  the  quantity  whose  maximum  and 
minimum  values  are  required  the  following  expression  : 

1       2b      a2  -  b2 


2  s       > 

tAs  m/  U/ 


neglecting  a  constant  multiplier. 

Accordingly,  the  solutions  of  the  problem  are — (1)  the 

maximum  and  minimum  values  of  x,  i.e.  a  +  b  and  a  -  b. 

du 
(2)  the  solutions  of  the  equation  — ,  i.e.  of 

ax 

1       \b      3  (a2  -  b2)  _ 

xl      Xs  ar 

or  x2  +  ^bx  -  3  {a2  -  b2)  =  o ; 

whence  we  get      x  =  ystf  +  b2  -  2b, 

neglecting  the  negative  root,  which  is  inadmissible. 

Again,  if  b  >  a,  */$a2  +  b2  -  2b  is  negative,  and  accord- 
ingly in  this  case  the  solution  given  by  (2)  is  inadmissible. 

*  This  problem  occurs  in  Astronomy,  in  finding  "when  a  planet  appears 
brightest,  the  orbits  being  supposed  circular. 


Maxima  and  Minima  Values  of  an  Implicit  Function.     1 85 


If  a  >  b,  it  remains  to  see  whether  y^fl2  +  b2  -  2b  lies 
between  the  limits  a  +  b  and  a  -  b.  It  is  easily  seen  that 
V^tf2  +  b2  -  2bis>  a  -  b:  the  remaining  condition  requires 

a  +    b   >  Via*  +  b2  -  2b, 

or  a  +  36  >  y^a2  +  b2, 

or  «2  +  tab  +  gb2  >  3a2  +  b2, 

i.  e.  4&2  +  3«&  >  fl2, 

oa2     25a2  7     3&     5# 

10        10  44 

or,  finally,  b  >  -. 

We  see  accordingly  that  this  gives  no  real  solution  unless 
the  lesser  of  the  given  sides  exceeds  one-fourth  of  the 
greater. 

When  this  condition  is  fulfilled,  it  is  easily  seen  that  the 
corresponding  solution  is  a  maximum,  and  that  the  solutions 
corresponding  to  x  =  a  +  b,  and  x  =  a  -  b,  are  both  minima 
solutions. 

152.  Maxima  and  Minima  Values  of  an  Implicit 
Function. — Suppose  it  be  required  to  find  the  maxima  or 
minima  values  of  y  from  the  equation 

fix,  y)  =  o. 

Differentiating,  we  get 

du     du  dy 

dx     dy  dx       ' 

where  u  represents  f(x,  y).    But  the  maxima  and  minima 

du 
values  of  y  must  satisfy  the  equation  —-  =  o :  accordingly  the 


1 86    Maxima  and  Minima  of  Functions  of  a  Single  Variable. 

maximum  and  minimum  values  are  got  by  combining*  the 

..        du  n 

equations  -7-  =  o,  and  u  =  o. 
dx 

153.  maximum  and  Minimum  in  case  of  a  Func- 
tion of  two  dependent  Variables. — To  determine  the 
maximum  or  minimum  values  of  a  function  of  two  variables, 
x  and  y,  which  are  connected  by  a  relation  of  -the  form 

fix,  y)  =  o. 

Let  the  proposed  function,  0  (x,  y)  be  represented  by  u ; 
then,  by  Art.  101,  we  have 

d<p  df  dcj)  df 
du  dx  dy  dy  dx 
dx  df 

dy 

But  the  maxima  and  minima  values  of  u  satisfy  the 

du 
equation  —  =  o,  hence  the  values  of  x  and  y  derived  from 
ax 

the  equations /(#,  y)  =  o,  and 

dcj)  df     d<f>df 

dx  dy      dy  dx       ' 

furnish  the  solutions  required.     To  determine  whether  the 

solution  so  determined  is  a  maximum  or   a  minimum,   it 

d2u 
is  necessary  to  investigate  the  sign  of  — .     We  add  an 

ax 

example  for  illustration. 

154.  Given  the  four  sides  of  a  quadrilateral,  to  find  when  its 
area  is  a  maximum. 

Let  a,  b,  c}  d  be  the  lengths  of  the  sides,  0  the  angle 
between  a  and  b,  xfj  that  between  c  and  d.  Then  ab  sin  (j> 
+  cd  sin  ^  is  a  maximum ;  also 

a2  +  b2  -  2ab  cos  <j>  =  c2  +  d2  -  2cd  cos  \p 

being  each  equal  to  the  square  of  the  diagonal. 

*  This  result  is  evident  also  from  geometrical  considerations. 


Maximum  Quadrilateral  of  Given  Sides.  187 

Hence  ab  cos  6  +  cd  cos  \L  —  =  o 

r  d§ 

for  a  maximum  or  a  minimum ;  also, 

ab  sin  6  =  cd  sin.  \p-f- ; 

.*.  tan  <f>  +  tan  ip  =  o,  or  <£  +  \p  =  1 8o°. 

Hence  the  quadrilateral  is  inscribable  in  a  circle. 

That  the  solution  arrived  at  is  a  maximum  is  evident 
from  geometrical  considerations ;  it  can  also  be  proved  to  be 
so  by  aid  of  the  preceding  principles. 

For,  substitute  —=—. — ^7  instead  of  -?-.  and  we  get 
cd  sin  \p  d(j> 

du  _  ab  sin  (0  +  \p) 
d(p  sin  ip 

„  d2u       ab  cos  (d>  +  \L)  f        d\L\  .  .  . 

Hence   -r-z  = ^~ — —    1  +  -f- 1  +  a  term  which 

d<p"  sin  \p         \         d(pj 

d  u 
vanishes  when  6  +  xp  =  1 8o° ;  and  the  value  of  —  becomes 

dty 

in  this  case 

ab    f       ab 
1  + 


sin  ip\*  '  cdj 
which  being  negative,  the  solution  is  a  maximum. 


1 88  Examples. 

Examples. 

3/* 

1.  Prove  that  a  sec  9  +  b  cosec  0  is  a  minimum  when  tan  6  =  A/-. 

\  a 

2.  Find  when  4#3  —  15a;3  +  12*  -  1  is  a  maximum  or  a  minimum. 

Am.  x  =  |r,  a  max. ;  cc  =  2,  a  min. 

3.  If  «  and  5  be  such  that  f{d)  =  f(b),  show  that  f(x)  has,  in  general,  a 
maximum  or  a  minimum  value  for  some  value  of  x  between  a  and  b. 

4.  Find  the  value  of  x  which  makes 

sin  x  .  cos  x 


cos2(6o°  —  x) 
a  maximum.  Am.  x  =  300. 

<r.  If  -\1 — ^-r  be  a  maximum,  show  immediately  that  "^-r  is  a  minimum. 

sin2* 

6.  Find  the  value  of  cos  x  when  is  a  maximum. 

V5  —  4  cos  a; 


-4«s.  cos* 


=  5-A/i3 


1+3*     .  .  12 

7.  Find  when  —  is  a  maximum.  „    *  =  -7 . 

V^4  +  5^2  •* 

a;2  +  ax  +  5 

8.  Apply  the  method  of  Ex.  5  to  the  expression -=. 

rtr  J  x*  —  ax  +  0 

9.  What  are  the  values  of  *  which  make  the  expression 

2xz  -  2 1*2  4-  36*  —  20 

a  maximum  or  a  minimum  ?  and  (2)  what  are  the  maximum  and  minimum 
values  of  the  expression  ?  -dns.  x  =  1,  a  max.  ;  *  =  6,  a  min. 

.  ma 

«o.  u  =  xm(a  -  xy.  -Ans.  x  =  -^j—^  a  maximuEl- 

11.  Given  the  angle  C  of  a  triangle  ;  prove  that  sin2^  +  sin2j?  is  a  maximum, 
and  cos2-4  +  cos2i?  a  minimum,  when  A  =  B. 

12.  Find  the  least  value  of  ae*x  +  be-**.  Ans.  2\/ab, 
J                  (a  -  x)(b  -  x) 


r 


Examples.  189 

14.  Show  that  b  +  c  (x  -  a)%,  when  x  =  «,  is  a  minimum  or  a  maximum 
according  as  c  is  positive  or  negative. 

15.  u  =  xcosx.  Avis,  x  =  cot  x. 

16.  Prove  that  xx  is  a  maximum  when  x  =  e. 

%    /Wi 

1 7.  TanOT#  .  tanw  (a  —  x)  is  a  maximum  when  tan  (a  —  ix)  =  tan  a  ? 

x         y  w  +  m 

18.  Prove  that : is  a  minimum  when  x  =  e. 

logx 

19.  Given  the  vertical  angle  of  a  triangle  and  its  area,  find  when  its  hase  is 
a  minimum. 

20.  Given  one  angle  A  of  a,  right-angled  spherical  triangle,  find  when  the 
difference  betweeen  the  sides  which  contain  it  is  a  maximum. 

do 
Here  tan  e  cos  A  =  tan  b ;  and  since  c  —  b  is  a  maximum,  — ■  =  i . 

db 

Hence  we  find  tan  b  =  y  cos  A. 

This  question  admits  of  another  easy  solution ;  for,  as  in  Art.  112,  we  have 

sin  (c-b)  „A 

•— ; ~  =  tan-  — ; 

sin  (e  +  b)  2 

consequently  sin  (e  -  b)  becomes  a  maximum  along  with  sin  (c  +  b),  since  A  is 
constant ;  and  hence  c  —  b  is  a  maximum  when  e  4-  b  =  900. 

This  problem  occurs  in  Astronomy,  in  finding  when  the  part  of  the  equation 
of  time  which  arises  from  the  obliquity  of  the  ecliptic  is  a  maximum. 

21.  Prove  that  the  problem,  to  describe  a  circle  with  its  centre  on  the 
circumference  of  a  given  circle,  so  that  the  length  of  the  arc  intercepted  within 
the  given  circle  shall  be  a  maximum,  is  reducible  to  the  solution  of  the  equation 
0  =  cot  6. 

22.  A  perpendicular  is  let  fall  from  the  centre  on  a  tangent  to  an  ellipse, 
find  when  the  intercept  between  the  point  of  contact  and  the  foot  of  the  perpen- 
dicular is  a  maximum.   Prove  that  p  =  y  ab,  and  intercept  =  a  —  b. 

23.  A  semicircle  is  described  on  the  axis-major  of  an  ellipse ;  draw  a  line  from 
one  extremity  of  the  axis  so  that  the  portion  intercepted  between  the  circle  and 
the  ellipse  shall  be  a  maximum. 

24.  Draw  two  conjugate  diameters  of  an  ellipse,  so  that  the  sum  of  the 
perpendiculars  from  their  extremities  on  the  axis-major  shall  be  a  maximum. 

25.  Through  a  point  0  on  the  produced  diameter  AB  of  a  semicircle  draw  a 
secant  ORB,',  so  that  the  quadrilateral  ABBB!  inscribed  in  the  semicircle  shall 
be  a  maximum. 

Prove  that,  in  this  case,  the  projection  of  BR'  on  AB  is  equal  in  length  to 
the  radius  of  the  circle. 

26.  If  sin  <p  =  k  sin  \p,  and  ty  +  i//  =  a,  where  a  and  h  are  constants,  prove 
that  cos  rp'  cos  <p  is  a  maximum  when  tan3<£  =  tan  ty  tan  t//\ 


190  Examples, 

27.  Find  the  area  of  the  ellipse 

ax2  +  zhxy  +  by2  =  c 
in  terms  of  the  coefficients  in  its  equation,  by  the  method  of  Art.  146. 

(1)  for  rectangular  axes.  Ans.  —  . 

y  ab  —  h2 

ire  sin  o> 


(2)  for  oblique. 


V  ab  —  h2 


28.  A  triangle  inscribed  in  a  given  circle  has  its  base  parallel  to  a  given  line, 
and  its  vertex  at  a  given  point ;  find  an  expression  for  the  cosine  of  its  vertical 
angle  when  the  area  is  a  maximum. 

29.  Find  when  the  base  of  a  triangle  is  a  minimum,  being  given  the  ver- 
tical angle  and  the  ratio  of  one  side  to  the  difference  between  the  other  and  a 
fixed  line. 

30.  Of  all  spherical  triangles  of  equal  area,  that  of  the  least  perimeter  is 
equilateral  ? 

31.  Let  uz  +  xz  —  $axu  =  o ;  determine  whether  the  value  x  =  o  gives  u  a 
maximum  or  minimum.  Ans.  Neither. 

32.  Show  that  the  maximum  and  minimum  values  of  the  cubic  expression 

ax3  +  $bx2  -f  $cx  +  d 
are  the  roots  of  the  quadratic 

ah2-  2Gz-  A  =  0; 
where  G  =  a2d  —  yibc  +  2bs,  and  A  =  a2d2  +  qac3  +  qdbz  —  $b2c2  —  6abcd. 

33.  Through  a  fixed  point  within  a  given  angle  draw  a  line  so  that  the 
triangle  formed  shall  be  a  minimum. 

The  line  is  bisected  in  the  given  point. 

34.  Prove  in  general  that  the  chord  drawn  through  a  given  point  so  as  to 
cut  off  the  minimum  area  from  a  given  curve  is  bisected  at  that  point. 

35.  If  the  portion,  AB,  of  the  tangent  to  a  given  curve  intercepted  by  two 
fixed  lines  OA,  OB,  be  a  minimum,  prove  that  PA  =  NB,  where  JP  is  the  point 
of  contact  of  the  tangent,  and  N  the  foot  of  the  perpendicular  let  fall  on  the 
tangent  from  0. 

36.  The  portion  of  the  tangent  to  an  ellipse  intercepted  between  the  axes  is 
a  minimum :  find  its  length.  Ans.  a  +  b. 

37.  Prove  that  the  maximum  and  minimum  values  of  the  expression,  Art.  147, 
are  roots  of  the  biquadratic 

(a  —  ua')2  (d  -  ud')2  +  4  (a  —  ua')  (c  —  uc'}3  +  4  (d  -  ud')  (b  -  ub')z 

—  3  (b  -  ub')2 (c  —  uc')2  —  6 (a  —  ua)  (b  —  ub')  (c  —  uc)  (d -  ud')  =  o. 


(     i9i     ) 


CHAPTEE  X. 


MAXIMA     AND     MINIMA    OF     FUNCTIONS    OF    TWO    OR   MORE   IN- 
DEPENDENT VARIABLES. 

155.  Maxima   and  Minima   for   Two  Variables. — In 

accordance  with  the  principles  established  in  the  preceding 
chapter,  if  $  (%,  y)  be  a  maximum  for  the  particular  values 
x0  and  y0,  of  the  independent  variables  x  and  y,  then  for  all 
small  positive  or  negative  values  of  h  and  kf  <p  (x0,  y0)  must 
be  greater  than  <j>  (x0  +  h,y0  +  h);  and  for  a  minimum  it  must 
be  less. 

Again,  since  x  and  y  are  independent,  w$  may  suppose 
either  of  them  to  vary,  the  other  remaining  constant; 
accordingly,  as  in  Art.  138,  it  is  necessary  for  a  maximum 
or  minimum  value  that 

du  _  du 

^  =  o,  and-  =  o;  (,) 

omitting  the  case  where  either  of  these  functions  becomes 
infinite. 

156.  Lagrange's  Condition. — We  now  proceed  to 
consider  whether  the  values  found  by  this  process  correspond 
to  real  maxima  or  minima,  or  not. 

Suppose  x0,  y0  to  be  values  of  x  and  y  which  satisfy  the 

equations 

du  .  du 

-7-  =  o,  and  -7-  =  o, 

ax  dy 

and  let  A,  B,  C  be  the  values  which  — ,  — — ,  — -  assume 

dx*    dxdy  dy% 

when  %0  and  y0  are  substituted  for  x  and  y ;  then  we  shall 

have 

$(#<>+  h, yo  +  k)- (j>(xQ, y0)= (Ah?+  zBhk+Ck2)  +  &c.  (2) 


192     Max.  and  Min.for  two  or  more  Independent  Variables, 

But  when  h  and  k  are  very  small,  the  remainder  of  the 
expansion  becomes  in  general  very  small  in  comparison  with 
the  quantity  AW  +  zBhk  +  Ck2 ;  accordingly  the  sign  of 
<j>(x0  +  h,  y0  +  k)  -  <p(x0)  y0)  depends  on  that  of 

a**        bit      mi-        AAh  +  Bk)2  +  k2(AC-B2) 

Ah2  +  2Bhk  +  Ck2,  i.e.  of - — - — -y  • 

A 

Now,  in  order  that  this  expression  should  be  either  always 
positive  or  always  negative  for  all  small  values  of  h  and  k, 
it  is  necessary  that  AC  -  B2  should  not  be  negative;  as,  if 
it  be  negative,  the  numerator  in  the  preceding  expression 
would  be  positive  when^=o,  and  negative  when  Ah  +  Bk  =  o. 
Hence,  the  condition  for  a  real  maximum  or  minimum  is 
that  AC  should  not  be  less  than  B2,  or 

d2u  d2u  (  d2u  \2e 

dec2  dy%  \dxdyj  ' 

and,  when  this  condition  is  satisfied,  the  solution  is  a  maxi- 
mum or  a  minimum  value  of  the  function  according  as  the 
sign  of  A  is  negative  or  positive. 

If  B2  be  >  AC  the  solution  is  neither  a  maximum  nor  a 
minimum. 

The  necessity  of  the  preceding  condition  was  first  estab- 
lished by  Lagrange  ;*  by  whom  also  the  corresponding  con- 
ditions in  the  case  of  a  function  of  any  number  of  variables 
were  first  discussed. 

Again,  if  A  =  o,  B  =  o,  C  =  o,  then  for  a  real  maximum 
or  minimum  it  is  necessary  that  all  the  terms  of  the  third 
degree  in  h  and  k  in  expansion  (2)  should  vanish  at  the  same 
time,  while  the  quantity  of  the  fourth  degree  in  h  and  k 
should  preserve  the  same  sign  for  all  values  of  these  quan- 
tities.    See  Art.  138. 

The  spirit  of  the  method,  as  well  as  the  processes  em- 
ployed in  its  application,  will  be  illustrated  by  the  following 
examples. 

157.  To  find  the  position  of  the  point  the  sum  of  the 
squares  of  whose  distances  from  n  given  points  situated  in 
the  same  plane  shall  be  a  minimum. 

*  Theorie  des  Fonctions.     Deuxieme  Partie.  Ch.  onzieme. 


Maxima  and  Minima  for  Two  or  more  Variables.        193 

Let  the  co-ordinates  of  the  given  points  referred  to 
rectangular  axes  be 

(«i,  &i),  (a2,  b2),  (a3f  h)  .  .  .  (an,  bn),  respectively; 

(x,  y)  those  of  the  point  required ;  then  we  have 

u  =  (x  -  ax)2  +  (y  -  bxy  +  (x  -  a2)2  +  (y  -  b2)2  +  .  .  a 

■    +  (x  -  an)2  +  (y  -  bny 
a  minimum ; 

du  ,  . 

.*.  —  =  x-a1  +  x~o2  + . . . -1- x - an  =  nx-[ai+a2+ . . ,+an) -o; 
dx  v  / 

ftlH 

-r=y-bi  +  y-b2+.  ,.+y-bn=ny-(bl+b2+...  +  bn)  =  o. 

tt                      ax  +  a2  +  .  .  .  +  an  bi  +  b2  +  .  .  .  +  bn 

Hence       x  = ,    y 


n  n 

and  the  point  required  is  the  centre  of  mean  position  of  the 
n  given  points. 

From  the  nature  of  the  problem  it  is  evident  that  this 
result  corresponds  to  a  minimum. 

This  can  also  be  established  by  aid  of  Lagrange's  con- 
dition, for  we  have 

_  <Fu  _  n  _  dhi   __  n  _  d*u  _ 

dx%  dxdy       '  dy% 

In  this  case  AC  -  B*  is  positive,  and  A  also  positive; 
and  accordingly  the  result  is  a  minimum. 

158.  To  find  the  Maximum  or  Minimum  Value 
of  the  expression 

ax2  +  by2,  +  2hxy  +  2gx  +  2fy  +  c. 

Denoting  the  expression  by  u,  we  have 

1  du 

--  =  ax  +  hy  +  g  =  o, 

i  du     _       ,  i 

-  —  -hx  +  by  +f-  o. 
2dy 


194       Maxima  and  Minima  for  Two  or  more  Variables. 

Multiplying  the  first  equation  by  x,  the  second  by  y,  and 
subtracting  their  sum  from  the  given  expression,  we  get 

u  =  gx+fy  +  c; 

whence,  eliminating  x  and  y  between  the  three  equations, 
we  obtain 

a    h    g 

u(ah  -  h2)  =      h    b    f    .  (3) 

9    f    c 

This  result  may  also  be  written  in  the  form 

dA 

where  A  denotes  the  discriminant  of  the  proposed  expression. 

.      .  d2u  d2u       ,         d2u 

Agam,       -  =  2«,      -.2b,      _**. 

Hence,  if  ab  -  h2  be  positive,  the  foregoing  value  of  u  is  a 
maximum  or  a  minimum  according  as  the  sign  of  a  is  negative 
or  positive. 

If  h2  >  ab,  the  solution  is  neither  a  maximum  nor  a 
minimum. 

The  geometrical  interpretation  of  the  preceding  result  is 
evident ;  viz.,  if  the  co-ordinates  of  the  centre  be  substituted 
for  x  and  y  in  the  equation  of  a  conic,  u  =  o,  the  resulting 
value  of  u  is  either  a  maximum  or  a  minimum  if  the  curve 
be  an  ellipse,  but  is  neither  a  maximum  nor  a  minimum  for 
a  hyperbola ;  as  is  also  evident  from  other  considerations. 

159.  To  find  the  Maxima  and  Minima  Values 
of  the  Fraction 

ax2  +  by2  +  2hxy  +  2gx  +  ify  +  c 
dx2jr  bfy2+  2h'xy+  2g'x+2f'y+c'' 

Let  the  numerator  and  denominator  be  represented  by 
<pi  and  02 ;  then,  denoting  the  fraction  by  u,  we  get 

0i  =  Ufa.  (a) 


Examples  for  Two  Variables.  195 

Differentiate  with  respect  to  x  and  y  separately,  then 

d6i      du  d(b2     dd)i      du  d<b2 

dx      dxr  dx       dy      dy r  dy 

but  for  a  maximum  or  a  minimum  we  must  have 


du 
dx 


du 


hence,  the  required  solutions  are  given  by  the  equations 

ax  +  hy  +  g  =  u{a'x  +  tiy  +  g'), 

hx  +  by  +f  =  u{Jix  +  Vy  +/'). 

Multiplying  the  former  by  x,  the  latter  by  y,  and  subtracting 
the  sum  from  the  equation  (a),  we  get 

gx  +fy  +  c=  u(gx  +fy  +  c). 
These  equations  may  be  written 

{a  -  du)x  +  (h  -  h'u)y  +  g  -  g'u  -  o, 
(h  -  h'ii)x  +  (b  -  b'u)y  +f  -fu  =  o, 
(g  -  g'u)x  +  (f-fu)y  +  c-  cu  =  o. 

Eliminating  x  and  y,  we  get  the  determinant 
a  -  a'u     h  —  h'u     g  -  g'u 
h  -  h'u     b  -  b'u    f  -fu     =  o.  (4) 

g  -  g'u     f-fu     c  -  c'u 

The  roots  of  this  cubic  equation  in  u  are  the  maxima  and 
minima  required. 

This  cubic  is  the  same  as  that  which  gives  the  three 
systems  of  right  lines  that  pass  through  the  points  of 
intersection  of  the  conies  (pi  =  o,  $2  =  o* 


*  Salmon's  Conic  Sections,  Art.  370. 
02 


196      Maxima  and  Minima  for  Tivo  or  more  Variables. 

The  cubic  is  written  by  Dr.  Salmon  in  the  form 

AV  +  0V  +  eu  +  A  =  o,  (5) 

where  A,  A'  denote  the  discriminants  of  the  expressions  0i  and 
02,  and  0,  0'  are  their  two  other  invariants. 

On  the  proof  of  the  property  that  the  coefficients  are  in- 
variants compare  Art.  144. 

The  cubic  reduces  to  a  quadratic  if  either  the  numerator 
or  the  denominator  be  resolvable  into  linear  factors ;  for  in 
this  case  either  A  =  o,  or  A'  =  o. 

If  both  the  numerator  and  denominator  be  resolvable  into 
factors,  the  cubic  reduces  to  the  linear  equation 

e'u  +  0  =  o, 

and  has  but  one  solution,  as  is  evident  also  geometrically. 

160.  To  find  the  Maxima  or  Minima  Values  of 
x2  +  y2  +  s2,  "where 

ax2  +  by2  +  cz2  +  2hxy  +  2gxz  +  2fzy  =  1. 

x         1/ 
Let  u  =  x2  +  y2  +  z2 ;  substitute  of  and  yf  for  -  and  -,  and 

z  z 

we  have 

of2  +  y'2  +  1 
u  = 


=  o.  (6) 


ax'2  +  byf2  +  c  +  2hxf/tf  +  2gx'  +  2fy'* 

Accordingly  the  cubic  of  formula  (4)  becomes  in  this  case 

a  -  ic1        h  g 

h        b-  u~x     f 

g  f      c-if 

This  is  the  well-known  cubic*  for  determining  the  axes  of 
a  surface  of  the  second  degree  in  terms  of  the  coefficients  in 
its  equation  :  when  expanded  it  becomes 

u~z  -  {a  +  b  +  c)u~2  +  (ab  +  be  +  ac  -  f2  -  g2  -  h2)wl 
+  (a/2  +  bg2  +  eh2  -  abe  -  2fgh)  =  o. 

*  See  Salmon's  Geometry  of  Three  Dimensions,  3rd  ed.,  Art.  82. 


Application  of  Lagrange's  Theorem.  197 

161.    Application    of    Lagrange's    Condition. — In 

applying  this  condition  to  the  general  case  of  Art.  159,  we 
write  the  equation  in  the  form 

from  which  we  get,  on  making  —  =  o,  and  —  =  o, 

ax  ay 

d2(j>i        d2<j)2  d2u 

dx2  dx2       ^2  dx2' 

d2^!         d2(j>2  d2u 

dxdy        dxdy        dxdy' 

d26\        d2<b2  d2u 


but 


dy2  dy2       r  dy2 

d2(j>i  =  (Pfc  _      ,      d2(jn 

dx2        "  '     dxz  '     dxdy         ' 


Hence 


i 


**  IS w -©)!  ■ 4{(a  -  °'w)(J  - 6%)  - (*  -  *w  ' 

Accordingly,  the  sign  of  AC  -  B2  is  the  same  as  that  of 
the  quadratic  expression 

(ab  -  h2)  -  (aV  +  Id  -  2hh')  u  +  (dbf  -  h'2)u\  (7) 

where  u  is  a  root  of  the  cubic  (4)  or  (5). 

If  A2  represent  the  determinant  in  (4),  the  preceding 

quadratic  expression  may  be  written  in  the  form  — — 2. 

Again,  ul9  u2,  u3  representing  the  roots  of  the  cubic  (4)  ; 
a,  |3,  those  of  the  quadratic  (7)  ;  if  ux  be  a  real  maximum  or 
minimum  value  of  u,  we  must  have  (ux  -  a)(ui  -  ff)(dbf  -  h'2) 
a  positive  quantity. 

Accordingly,  if  dbr  -  h'z  be  positive,  ux  must  not  lie  be- 
tween the  values  a  and  /3.     Similarly  for  the  other  roots. 


198       Maxima  and  Minima  for  Two  or  more  Variables. 

If  all  the  roots  of  the  cubic  lie  outside  the  limits  a  and  ]3, 
they  correspond  to  real  maxima  or  minima,  but  any  root 
which  lies  between  a  and  j3  gives  no  maximum  or  minimum. 

In  the  particular  case  discussed  in  Art.  1 60  the  roots  of 
the  cubic  (6)  are  all  real,  and  those  of  the  quadratic 

a  -  u~l,        h 

=  o  are  interposed  between  the  roots  of  the 
h,      b  -  w1 

cubic.  (See  Salmon's  Higher  Algebra,  Art.  44).  Accord- 
ingly, in  this  case  the  two  extreme  roots  furnish  real  maxima 
and  minima  solutions,  while  the  intermediate  root  gives 
neither.  This  agrees  with  what  might  have  been  anticipated 
from  the  properties  of  the  Ellipsoid ;  viz.,  the  axes  a  and  c 
are  real  maximum  and  minimum  distances  from  the  centre  to 
the  surface,  while  the  mean  axis  b  is  neither. 

It  would  be  unsuited  to  the  elementary  nature  of  this 
treatise  to  enter  into  further  details  on  the  subject  here. 

162.  Maxima  or  Minima  of  Functions  of  three 
Variables. — Next,  let  u  =  <p(x,  y,  z),  and  suppose  x0,  y0,  s0 
to  be  values  of  x,  y,  z,  which  render  u  a  maximum  or  a  mini- 
mum ;  then,  if  x,  y,  z  be  independent  of  each  other,  by  the 
same  reasoning  as  before,  it  is  obvious  that  xw  y0  ,  20  must 
satisfy  the  three  equations 

du  du  du 

dx       '      dy       '     dz        9 

omitting  the  case  of  infinite  values. 
Accordingly  we  must  have 

A2  JP  P 

(j>(x0  +  h,  yQ  +  h,  z0  +  l)  -  <j>{x0,y0iZo)=A—  +B— ■  +  C— — 

+  FM+  Ghl  +  mk  +  &e. 
where  A,  B,  C,  F,  G,  H,  are  the  values  that 

d2u     d2u     d2u       d2u        d2u        d2u 


dx%1     dy29     dz2'     dydz      dxdz*     dxdy 

respectively  assume  when  x0,  y0,  Zo  are  substituted  for  x,  y,  2 
in  them. 


Maxima  or  Minima  for  Three  Variables.  199 

Now,  in  this,  as  in  the  case  of  two  independent  variables, 
it  is  necessary  for  a  real  maximum  or  minimum  value  that 
the  preceding  quadratic  function  should  be  either  always 
positive  or  always  negative  for  all  small  real  values  of  A,  k, 
and  I. 

Substituting  al  for  h,  and  (51  for  k,  and  suppressing  the 
positive  factor  P,  the  expression  becomes 

Aa2  +  B(52+  C+2F(3  +  2Ga  +  2Ha[3,  (8) 

(sp  +  Gy 


or 


a"  +  2a 


A 


+  B(52  +  2F(5  +  C. 


Completing  the  square  in  the  first  term,  and  multiplying  by 
A,  we  get 

(Aa  +  S(5  +  G)2  +  (AB-S2)(52  +  2{AF-  GH)(5  +{AC-  G2). 

Moreover,  since  the  first  term  is  a  perfect  square,  in  order 
that  the  expression  should  preserve  the  same  sign,  it  is  neces- 
sary that  the  quadratic 

(AB  -  S2)(52  +  2(AF-  OH")j3  +  AC  -  G% 

should  be  positive  for  all  values  of  j3  :  hence  we  must  have 

AB-R2>  o,  (9) 

and  (AB  -  E*)(AC  -  (?)  >  (AF-  GE)2, 

or         A(ABC  +  2FGK  -  AF2  -  BG2  -  CH2)  >  o,        (10) 

i.e.  A  and  A  must  have  the  same  sign,  A  denoting  the  dis- 
criminant of  the  quadratic  expression  (8),  as  before. 

Accordingly,  the  conditions  (9)  and  (10)  are  necessary 
that  x0,  Vq,  So  should  correspond  to  a  real  maximum  or  mini- 
mum value  of  the  function  u. 

When  these  conditions  are  fulfilled,  if  the  sign  of  A  be 
positive,  the  function  in  (8)  is  also  positive,  and  the  solution 
is  a  minimum  ;  if  A  be  negative,  the  solution  is  a  maximum. 

163.  Maxima  and  Minima  for  any  number  of 
Tariables. — The  preceding  theory  admits  of  easy  extension 


200      Maxima  and  Minima  for  Two  or  more  Variables. 


to  functions  of  any  number  of  independent  variables.  The 
values  which  give  maxima  and  minima  in  that  case  are  got 
by  equating  to  zero  the  partial  derived  functions  for  each 
variable  separately,  and  the  quadratic  function  in  the  ex- 
pansion must  preserve  the  same  sign  for  all  values ;  i.e.  it 
must  be  equivalent  to  a  number  of  squares,  multiplied  by 
constant  coefficients,  having  each  the  same  sign. 

The  number  of  independent  conditions  to  be  fulfilled  in  the 
case  of  n  independent  variables  is  simply  w  -  i,  and  not  2n  —  i, 
as  stated  by  some  writers  on  the  Differential  Calculus.  A 
simple  and  general  investigation  of  these  conditions  will  be 
given  in  a  note  at  the  end  of  the  Book. 

164.  To  investigate  the  Maximum  or  Minimum 
Value  of  the  Expression 

ax2  +  by2  +  cz2  +  2hxy  +  zgzx  +  zfyz  +  ipx  +  2qy  +  2rz  +  d. 

Let  u  denote  the  function  in  question,  then  for  its  maxi- 
mum or  minimum  value  we  have 

du        ,  s 

-—  =  2  {ax  +  hy  +  gz  +p)  =  o, 

—  =  2{hx  +  by  +fz  +  q)  =  o, 

—  =  2(gx  +fy  +  cz  +  r)  =  o ; 

hence,  adopting  the  method  of  Art.  158,  we  get 

u  =  px  +  qy  +  rz  +  d. 
Eliminating  x,  y,  z  between  these  four  equations,  we  obtain 
a     h    g    p 


h     b    f     q 

g     f    c     r 
p     q     r     d 


u 


a    h    g 
h     b    f 

g   f   c 


A      .       .        d2u  d2u       7    p 

Again,  since  —  =  2a,    —  =  2b,  &c, 


Maxima  or  Minima  for  two  or  more  Variables.  201 

the  result  is  neither  a  maximum  nor  a  minimum  unless 

a    h    g 

is  positive,  and  |  h    b    f 

9   f    0 


a    h 
h    b 


has  the  same  sign  as  a. 


The  student  who  is  acquainted  with  the  theory  of  surfaces 
of  the  second  degree  will  find  no  difficulty  in  giving  the 
geometrical  interpretation  of  the  preceding  result. 

165.  To  find  a  point  snch  that  the  sum  of  the 
squares  of  its  distances  from  n  given  points  shall  be 
a  Minimum. — Let  (a,  b,  c),  (of,  b\  c'),  &o.,  be  the  co-ordi- 
nates of  the  given  points  referred  to  rectangular  axes ;  x,  y,  z, 
the  co-ordinates  of  the  required  point ;  then 

(x  -  af  +  (y-  b)2  +  (z  -  c)2 

is  equal  to  the  square  of  the  distance  between  the  points 
[a,  b,  c),  and  (x,  yf  z). 
Hence 

u  =  (x  -  a)2  +  (y  -  by  +  (z  -  cf  +  (x  -  aj  +  {y  -  bj  +  (*  -  c)'2 

+  &c.  =  2(0  -  a)2  +  S(y  -  b)2  +  2(*  -  c)\ 

where  the  summation  is  extended  to  each  of  the  n  points. 
For  the  maximum  or  minimum  value,  we  have 

-=-  =  22(#  —  a)  =  2nx  -  2z,a  =  o, 
ax 

—  =  2^(y  -  b)  =  my  -  22&  =  o, 

is 

-=-  =  22vs  -  c)  -  2nz  -  22c  =  o ; 
d% 

2fl  2&  2c 

n       9        n  n 

i.e.  %0,  yoi  So  are  the  co-ordinates  of  the  centre  of  mean  posi- 


202         Maxima  and  Minima  of  Independent  Variables. 

tion  of  the  given  points.     This  is  an  extension  of  the  result 
established  in  Art.  157. 

A      .         d2u  d2u  d?u  d2u 

Agam      -  =  2n,  —  =  zn,  ^  =  zn,  —  =  o,  &c. 

The  expressions  (10)  and  (11)  are  both  positive  in  this  case, 
and  hence  the  solution  is  a  minimum. 

It  may  be  observed  with  reference  to  examples  of  maxima 
and  minima,  that  in  most  cases  the  circumstances  of  the  prob- 
lem indicate  whether  the  solution  is  a  maximum,  a  minimum, 
or  neither,  and  accordingly  enable  us  to  dispense  with  the 
labour  of  investigating  Lagrange's  conditions. 


Examples.  203 

Examples. 
Find  the  maximum  and  minimum  values,  if  any  such  exist,  of 

ax  4-  by  +  c  c  ±  «/  a2  +  b2  +  t? 

I.  — — — -— — .  Ans.  -  -  . 

%   +  y£  +  1 


ax  +  by  +  c 


„  ±     \A2  +  b2  +  c*. 


y  x2  +  y2  + 1 

3.  z*  +  y*  -  x2  +  xy  -  y2. 

(a),  x  —  o,  y  =  o,  a  maximum. 

03).  x  =  y  =  +  -,  a  minimum. 
—  2 

/  v  v  3  •  • 

(<y).  iu  =  -  y  =  + ?  a  minimum. 

4.  ##2  +  bxy  +  dz2  +  Ixz  +  myz. 

a;  =  y  =  2  =  o,  neither  a  maximum  nor  a  minimum. 

a  a       , 

5.  If  w  =  «#3y2  -  z4?/2  -  x3y3,  prove  that  x  -  -,  y  =  -  makes  w  a  maximum. 

2  3 

6.  Prove  that  the  value  of  the  minimum  found  in  Art.  165  is  the  -th  part  of 

n 

the  sum  of  the  squares  of  the  mutual  distances  between  the  n  points,  taken  two 
and  two. 

7.  Find  the  maximum  value  of 

-a2;tr2-/32j/2-y232  /  i  (a2         Tz        C2\ 

(ax  +  by±cz)e  .       Ans.  ^-  [^  +  ^  +  ^  j  . 

8.  Find  the  values  of  x  and  y  for  which  the  expression 
(aix  +  fay  +  tfi)2  +  («2»  +  %  +  c2)2  +  .  .  .  +  (an%  +  bny  +  cn)2 

becomes  a  minimum. 


(       204       ) 


CHAPTEE  XI. 

METHOD  OP  UNDETERMINED  MULTIPLIERS  APPLIED  TO  THE 
INVESTIGATION  OP  MAXIMA  AND  MINIMA  IN  IMPLICIT 
FUNCTIONS. 

1 66.  Method  of  Undetermined  Multipliers. — In  many 
cases  of  maxima  and  minima  the  variables  which  enter  into 
the  function  are  not  independent  of  one  another,  but  are  con- 
nected by  certain  equations  of  conditioD. 

The  most  convenient  process  to  adopt  in  such  cases  is 
what  is  styled  the  method  of  undetermined*  multipliers.  "We 
shall  illustrate  this  process  by  considering  the  case  of  a  func- 
tion of  four  variables  which  are  connected  by  two  equations 
of  condition. 

Thus,  let  u  =  (p(xly  x2,  #3 j  %i), 

where  %X9  x2,  x3,  a?4  are  connected  by  the  equations 

Fifa,  #2?  #3?  #0  =  o,  F2(%1,  #2,  #3 ?  #4)  =  o.  (1) 

The  condition  for  a  maximum  or  a  minimum  value  of  u 
evidently  requires  the  equation 

d<h    7        dd>  -         dd>    _        dd>    _ 

-rf-  axi  +~-  dx2  +  -T-  dxz  +  -p-  dx±  =  o. 

axi  ax2  dxs  dx± 

Moreover,  the  differentials  are  also  connected  by  the  rela- 
tions 

dF1  .       dF,  7       dFx  ,       dF1  . 

— —  dxy  +  —  dx2  +  -=—  dxz  +  -=—  ^4  =  o, 

(M?j  W5?2  W#?3  W#?4 

■rf^i   _       <ZFa  _       dF2  _       <ZF8  _ 

■3-  a#i  +  — —  #ir2  +  -T-  «^3  +  -^ —  dXi  =  o. 

£a#1  &X%  CCX3  Ct/X^ 

Multiplying  the  first  of  the  two  latter  equations  by  the  arbitrary 


This  method  is  also  due  to  Lagrange.     See  Mec.  Anal.,  tome  1.,  p.  74. 


Method  of  Undetermined  Multipliers.  205 

quantity  Xi,  the  other  by  X2,  and  adding  their  sum  to  the  pre- 
ceding equation,  we  get 

d(j>      y    dF1     .     dF2\  '        fd(f>      .    dFx     .     dF2\ 
-~  +  Ai  — -  +  A2  —  )dxy  +    -^  +  Ai  —  +  A2  —  )dx2 
dxx  dxx  dxij  \ax2  dx2  ax2) 

(d<l>      _    dFx      _    dF2\  _       /<fy       .    rfJFl      ,   d[Fa\  _ 
+ hr^  +  Ai  -7-  +  a3  3-^3+   -T-+  Ai  3-  +  A2  —  kfc4=o. 
\ax3  dxs  dxz]  \dx±  ax^  ax±J 

As  Ai,  A2  are  completely  at  our  disposal,  we  may  suppose 
them  determined  so  as  to  make  the  coefficients  of  dxx  and  dx% 
vanish.     Then  we  shall  have 

d6      _    dFx      _    <fJPa\  7       /^0       x    ^1      a    dF£\  . 
ofo?3  gm?3  «a?3/  \a^4  «a?4  "^4/ 

Again,  since  we  may  regard  x3,  x±  as  independent  variables, 
and  a?j,  #2  as  dependent  on  them  in  consequence  of  the  equa- 
tions (1),  it  follows  that  the  coefficients  of  dx3  and  dx±  in  the 
last  .equation  must  be  separately  zero,  for  a  maximum  or  a 
minimum  ;  consequently,  we  must  have 

dd>       .    dFx      ^    dF2 
-£-  +  Ai  -=-  +  A2  —  =  o, 

ao?3  «a?3  aa?3 

■/-  +  Ai  -7-  +  A2  -J-  =  o. 

«#4  «#4  UXi 

These,  along  with  equations  (1)  and 

dd>       -    ^      .    ^ 
-^-  +  Ai  —  +  A2  -j-  =  o, 
dxi  dxx  dxx 

d6       .    dFx      ^    ^2 
-^-  +  Aj  —  +  A2  —  =  o9 

ax2  ax%  wXz 

are  theoretically  sufficient  to  determine  the  six  unknown 
quantities,  Xi,  x2,  x3,  a?4,  Ax,  A2 ;  and  thus  to  furnish  a  solution 
of  the  problem  in  general. 

This  method  is  especially  applicable  when  the  functions 
Fi9  F2,  &c,  are  homogeneous ;  for  if  we  multiply  the  preceding 


206  Method  of  Undetermined  Multipliers. 

differential  equations  by  xl9  x2,  x3,  x±,  respectively,  and  add, 
we  can  often  find  the  result  with  facility  by  aid  of  Euler's 
Theorem  of  Art.  103. 

There  is  no  difficulty  in  extending  the  method  of  undeter- 
mined multipliers  to  a  function  of  n  variables,  xx,  x2,  x3,  .  . ,. 
xn,  the  variables  being  connected  by  m  equations  of  condition. 

F1  =  o,Fz  =  o,F3  =  os  .  .  .  Fm  =  o, 

m  being  less  than  n ;  for  if  we  differentiate  as  before,  and 
multiply  the  differentials  of  the  equations  of  condition  by  the 
arbitrary  multipliers,  Ai,  A3,  .  .  .  \m  respectively;  by  the  same 
method  of  reasoning  as  that  given  above,  we  shall  have  the  n 
following  equations, 

dd>      .   dFi  .    dFm 

■j-  +  Ai—  -+...+  \m  -=—  =  o, 

axL         dxx  axi 

d(b  dFx  .     dFm 

—  +  Ai  —-  +  ...+  Am  -5 —  =  o, 
ax2         dx%  dx2 


d<p  dFx  .    dFm 

-j—  +  Ai  — f-  .  .  .  +  Am  — —  =  O. 

These,  combined  with  the  m  equations  of  condition,  are 
theoretically  sufficient  for  the  determination  of  the  m  +  n 
unknown  quantities 

Xxf    X<1)     •      •      •     Xny     Al,     A2)     •      •      •     A#}. 


Examples. 

1.  To  find  the  triangle  of  maximum  area  inscribed  in  a  given  circle. 
Let  R  denote  the  radius  of  the  circle,  A,  B,  C,  the  angles  of  an  inscribed 
triangle,  u  its  area  ;  then 

abe        _     .       .    .     _   .     _ 

u  =  — -  =  2.&2  sin  A  sin  B  sin  C. 
4H 

Also,  A  +  B+C=iSo°;     .'.  dA  +  dB  +  dC=o; 

and,  taking  logarithmic  differentials,  we  get 

cot  AdA  +  cot  BdB  +  cot  CdC=o, 


Examples.  207 

and  consequently 

tan  A  =  tan  B  =  tan  0;  hence  A  —  B  =  C  =  6o° ; 

and  therefore  the  triangle  is  equilateral. 

2.  Find  a  point  such  that  the  sum  of  the  squares  of  the  perpendiculars 
drawn  from  it  to  the  sides  of  a  given  triangle  shall  be  a  minimum. 

Let  x,  y,  z  denote  the  perpendiculars :  a,  b,  c  the  sides  of  the  triangle  ;  then 

u  =  x2  +  y2  +  z2  is  to  be  a  minimum ; 

also  ax  +  by  +  cz  =  double  the  area  of  a  triangle  =  2A  (suppose) ; 

.  *.  xdx  +  ydy  +  zdz  —  o,  adx  +  bdy  +  cdz  =  o  , 

.-.  x  =  Xa,  y  =  \b,  z  =  \c:  multiplying  these  equations  by  a,  b,  c,  respectively, 
and  adding,  we  obtain 

2A 

ax  +  by+cz  =  \  (a?  +  P  +  c2),  or      A  =  ga  +  ^  +  #\ 

zAa  2Ab  2A<? 

*'•   x  =  a%  +  b2  +  c*      V  =  a%  +  ¥  +  c2'    *  =  a2  +  b2  +  c2' 

which  determine  the  position  of  the  point.     The  minimum  sum  is  obviously 

4A2 
a2  +  ft  +  c*' 

3.  Similarly,  to  find  a  point  such  that  the  sum  of  the  squares  of  its  distances 
from  four  given  planes  shall  be  a  minimum.  Suppose  A,  B,  0,  D  to  represent 
the  areas  of  the  faces  of  the  tetrahedron  formed  by  the  four  planes ;  x}  y,  z,  w, 
the  perpendiculars  on  these  faces  respectively;  then,  as  in  the  preceding 
example,  we  have 

Ax  +  By+Cz  +  JDw  =  three  times  the  volume  of  the  tetrahedron  =  3  V (suppose), 

and  u  =  x2  +  y2  +  z2  +  to2,  a  minimum; 

.♦.  xdx  +  ydy  +  zdz  +  wdw  =  o, 

Adx  +  Bdy  +  Cdz  +  Ddw  =  o  ; 

hence  x  =  \A,  y  =  AjB,  z  =  xC,  w  =  KB ; 

9  V2 
and  proceeding  as  before,  we  get  «=  ^2        2  +  g2     J2» 

4.  To  prove  that  of  all  rectangular  parallelepipeds  of  the  same  volume  the  cube 
has  the  least  surface. 

Let  x,  y,  z  represent  the  lengths  of  the  edges  of  the  parallelepiped ;  then,  if 
A  denote  the  given  volume,  we  have 

xyz  =  A,  and  xy  +  xz  +  yz  a  minimum ; 

.*.  yzdx  +  xzdy  +  xydz  =  o, 

(y  +  z)  dx  +  (x  +  z)  dy  +  (x  +  y)dz  =  o ; 

hence  yz  -  \  (y  +  s),  xz  =  A  (x  +  z),  xy  =  A  (x  +  y) ; 

from  which  it  appears  immediately  that  x  =  y  =  z. 


208 


Method  of  Undetermined  Multipliers. 


167.    To      find     the      Maximum     and    Minimum 
Values  of 

ax2  +  by2  +  cz2  +  ihxy  +  2gzx  +  2fyz, 

where  the  variables  are  connected  by  the  equations 

Lx  +  My  +  Nz  =  o,  and  x2  +  y2  +  z2  =  1 . 

In  this  case  we  get  the  following  equations : 

ax  +  hy  +  gz  +  \XL  +  A2a?  =  o, 

hx  +  by  +fz  +  \XM  +  \2y  =  o, 

gx  +fy  +  cz  +  Aii^*  A2s  =  o. 

Multiply  the  first  by  x,  the  second  by  yy  the  third  by  z,  and 
add;  then 

u  +  A2  =  o,  or  A2  =  -  u. 

Hence  (a  -  u)  x  +  hy  +  gz  +  AiZ  =  o, 

hx  +  (b  -  u)  y  +fz  +  \M  =  o, 

gx  +fy  +  (c  -  u)  z  +  AiiV  =  o, 

Lx  +  My  +  iVs  =  o : 

eliminating  x,  y,  z  and  Ai,  we  get  the  determinant  equation 

a  -  u,      h,  g,      L 

h,        b-u,       f,      M 

g,  f,        c-u,   N 

L,         M,         N,       o 


=  o. 


« 


The  roots  of  this  quadratic  determine  the  maximum  and 
minimum  values  of  u. 

The  preceding  result  enables  us  to  determine  the  principal 
radii  of  curvature  at  a  given  point  on  a  surface  whose  equa- 
tion is  given  in  rectangular  co-ordinates. 


Application  to  Surfaces. 


209 


Again,  the  term  independent  of  u  in  this  determinant  is 
evidently 

a,    h,    g,    L 
h,     b,    f,    M 

g,  f,  c,  jst 

L,  M,  N,    o 

and  the  coefficient  of  u2  is  L2  +  M2  +  N2.  Accordingly,  the 
product  of  the  roots  of  the  quadratic  (2)  is  equal  to  the  frac- 
tion whose  numerator  is  the  latter  determinant,  and  denomi- 
nator L2  +  M2  +  N2.  From  this  can  be  immediately  deduced 
an  expression  for  the  measure  of  curvature*  at  any  point  on  a 
surface. 


*  Salmon's  Geometry  of  Three  Dimensions,  Art.  295. 


2IO 


Examples. 


Examples. 

1.  Find  the  minimum  value  of 

where  xi,  x%,  .  .  .  xn  are  subject  to  the  condition 

a\x\  +  awi  +  ...-(-  anxn  =  Jc.  Ans. 

2.  Find  the  maximum  value  of 

%p  yi  zr, 
where  the  variables  are  subject  to  the  condition 

ax  +  by  +  cz  =  p  +  q  +  r. 


& 


ar  +  a2  .  .  .  +  a2 


*»■  (WG)'- 


6  <p 

3.  If  tan  -  tan  -  =  m,  find  when  sin  d  —  m  sin  <$>  is  a  maximum. 

22  r 

4.  Find  the  maximum  value  of  (#  +  1)  («/  +  1)  (z  +  1)  where  ax  bv  cz  =  A. 

{log  (Aabe)}z 


Ans. 


27  log  as  .  log  5  .  log  c 


5.  Find  the  volume  of  the  greatest  rectangular  parallelepiped  inscribed  in 
the  ellipsoid  whose  equation  is 

x2      y2      z2  8  abc 

a2      £3      c2  3^/^ 

6.  Find  the  maximum  or  the  minimum  values  of  u,  being  given  that 

u  =  a2#2  +  62y2  +  c2z2,    x2  +  y2  +  z2  =  1,     and  fo?  +  my  +  wz  =  o. 
Proceeding  by  the  method  of  Art.  1 67,  we  get 

a2x  4-  Kx  +  fxl  =  o,     £2y  +  Ay  +  jum  =  0,     c3z  +  Az  +  /m  =  o. 
Again,  multiplying  by  x,  y,  z,  respectively,  and  adding,  we  get  A  =  -  u. 

.'.  (u  —  a2)  x  =  /xl,     (u  —  b2)  y  =  jxm,     (u  —  e2)  z  —  /j.n. 
Hence,  the  required  values  of  u  are  the  roots  of  the  quadratic 

I2  m2  n2 

+  ^  +■  =  o. 


u  -  a2      u  —  b'~      u  —  c- 


Examples.  211 

X2       ifl       z2 

7.  Given  —  +  —  +  —  =  I,  and  Ix  +  my  +  nz  =  o,  find  when  x2  +  y2  +  z2  is  a 
az      #2      c3 

maximum  or  minimum.  Proceeding,  as  in  the  last  example,  we  get  the  quadratic 

a2 12  b2m2  c2n2 

+ T5  + ~n  =  °. 


u  —  cfi      u  -  b2      u  —  c2 

This  question  can  be  at  once  reduced  to  the  last  by  substituting  in  our  equations 
ax,  by,  and  ez,  instead  of  x,  y,  z. 

8.  Of  all  triangular  pyramids  having  a  given  triangle  for  base,  and  a  given 
altitude  above  that  base,  find  that  whose  surface  is  least. 

Am.  Value  of  minimum  surface  is \/r2  +  p2,  where  a,  b,  c  repre- 

2 

sent  the  sides  of  the  triangular  base ;  r,  the  radius  of  its  inscribed  circle  ;  and  p, 
the  given  altitude. 

9.  Divide  the  quadrant  of  a  circle  into  three  parts,  such  that  the  sum  of  the 
products  of  the  sines  of  every  two  shall  be  a  maximum  or  a  minimum ;  and 
determine  which  it  is. 

10.  Of  all  polygons  of  a  given  number  of  sides  circumscribed  to  a  circle,  the 
regular  polygon  is  of  minimum  area?  For,  let  <pi,  <p%,  .  .  .  <pn  be  the  external 
angles  of  the  polygon,  then  the  area  can  be  easily  seen  to  be  in  general 


r2 

/           <t>l                  4>2 

tan  —  +  tan  2—  +  .  .  .  +  to 

V            2                     2 

where 

<pl  +  $2  •   •    •  +  <pn  =  27T, 

Hence,  for  a  minimum, 

01  =  02  =  <pZ  =..'.=  (prt 

<Pn\ 


11.  Of  all  polygons  of  a  given  number  of  sides  circumscribed  to  any  closed 
oval  curve  which  has  no  singular  points,  that  which  has  the  minimum  area 
touches  the  curve  at  the  middle  point  of  each  of  the  sides. 

12.  Given  the  ratio  sin  0 :  sin  ty,  and  the  angle  6,  find  when  the  ratio 
sin  (<p  +  6)  :  sin  ($  +  6)  is  a  maximum  or  a  minimum.  Am.  <p  +  ^  —  6. 

13.  Required  the  dimensions  of  an  open  cylindrical  vessel  of  given  capacity, 
so  that  the  smallest  possible  quantity  of  material  shall  be  employed  in  its  con- 
struction, the  thickness  of  the  base  and  sides  being  given. 

Am.  Its  altitude  must  be  equal  to  the  radius  of  its  base. 

14.  Show  how  to  determine  the  maximum  and  minimum  values  of  x2+y2+z2 
subject  to  the  conditions 

(*2  +  y2  +  z2)2  =  a2x2  +  b2y2  +  &z\ 

Ix  +  my  +  nz  =  o. 

r  2 


(  2I2  ) 


CHAPTEE  XII. 


TANGENTS   AND   NORMALS   TO   CURVES. 

1 68.  Equation  of  the  Tangent.— If  (x,  y),  [xx,  yx),  be  the 
co-ordinates  of  any  two  points,  P,  Q9  taken  on  a  curve,  and 


if  (X,  T)  be  any  point  on  the    Y 
line  which  joins  P  and  Q ;  then 
the  equation  of  the  line  PQ  is 


Y-y  =  (X-x) 


xx  -  x' 


Q 


n 


N     M 


X. 


in  which  X  and  T  represent  the    ° 

current  co-ordinates.  Flg*  8< 

If  now  the  point  Q  be  taken  infinitely  near  to  P,  the  line 
PQ  becomes  the  tangent  at  the  point  P,  and,  as  in  Art.  10, 
we  have  for  its  equation 


Y-y=(X-x) 


&y_ 
dx' 


(i) 


where  X,  Y  are  the  co-ordinates  of  any  point  on  the  line, 
and  x,  y  those  of  its  point  of  contact. 

For  example,  to  find  the  equation  of  the  tangent  to  the 


curve 


xnym  =  an+m. 


Taking  the  logarithmic  differentials  of  both  sides,  we  get 

n     mdy  dy        ny 

x     y  dx        '  dx       ma? 

and  the  equation  of  the  tangent  becomes 

nX     mY 

—  + =  m  +  n. 

x         y 


Tangents  Parallel  to  a  Given  Line.  213 

m  +  n 


If  we  make  X  =  o,  and  Y  =  o,  separately,  we  get 


n 


tn  +  n 
and y  for  the  lengths  of  the  intercepts  made  by  the 

tangent  on  the  axes  of  x  and  y,  respectively.  This  result 
furnishes  an  easy  geometrical  method  of  drawing  the  tangent 
at  any  point  on  a  curve  of  this  class. 

If  m  =  1,  n  =  1,  the  preceding  equation  represents  a 
hyperbola  ;  if  m  =  2,  and  n  =  -  1,  it  represents  a  parabola. 

169.  If  the  equation  of  the  curve  be  of  the  form 
f(x,  y)  =  o,  and  if  f(x,  y)  be  denoted  by  u,  we  have  from 
Art.  100, 

du 

dy  _  _dx 
dx        duf 
dy 

and  hence  the  equation  of  the  tangent  becomes 

t*-)s+(r-r>S-*  W 

The  points  on  the  curve  at  which  the  tangents  are 
parallel  to  the  axis  of  x  must  satisfy  the  equation  —  =  o ; 

QsX 

they  are  accordingly  given  by  the  intersection  of  the  curve, 

du 
u  =  o,  with  the  curve  whose  equation  is  —  =  o.     The  y  co- 

ax 

ordinates  at  such  points  are   evidently  in   general  either 

maxima  or  minima. 

Similar  remarks  apply  to  the  points  at  which  the  tangents 
are  parallel  to  the  axis  of  y. 

To  find  the  tangents  parallel  to  the  line  y  =  mx  +  n.  The 
points  of  contact  must  evidently  satisfy 

du         du 

—  +  m—  =0. 

dx         dy 

The  points  of  intersection  of  the  curve  represented  by 


214  Tangents  and  Normals  to  Curves. 

this  equation  with  the  given  curve  are  the  points  of  contact 
of  the  system  of  parallel  tangents  in  question. 

The  results  in  this  and  the  preceding  Article  evidently 
apply  to  oblique  as  well  as  to  rectangular  axes. 


Examples. 


i.  To  find  the  equation  of  the  tangent  to  the  ellipse 

x2      y2 
a2      o1 

du      2x      du      iy 

Here  Tx  =  #>    di  =  W 


and  the  required  equation  is 


!(z-*)  +  £(r-,)  =  o, 


xX      vY     x2 


0r  a2  +  £2       a2  +  bi      *' 

2.  Find  the  equation  of  the  tangent  at  any  point  on  the  curve 

xm        ym  Xxm-1         Yy»1-1 

— +  V-=i.  Ans. +  -f — =  i. 

3.  If  two  curves,  -whose  equations  are  denoted  by  u  =  o,  u'  —  o,  intersect  in 
a  point  (x,  y),  and  if  a  be  their  angle  of  intersection,  prove  that 

du  du'      du'  du 

dx  dy       dx  dy 
tan  w  =   T  ,   ■  /.. 
du  die      du  du 

dx  dx       dy  dy 

4.  Hence,  if  the  curves  intersect  at  right  angles,  we  must  have 

du  du'      du  du' 
dx  dx       dy  dy 

5.  Apply  this  to  find  the  condition  that  the  curves 

x2     y2  x2      y2 

a2  +  b2=h     aT2+bi2==l 

should  intersect  at  right  angles.  Ans.  a2  -  b2  =  a'2  -  b'2. 


Equation  of  Normal. 


215 


170.  Equation  of  formal. — Since  the  normal  at  any 
point  on  a  curve  is  perpendicular  to  the  tangent,  its  equation, 
when  the  co-ordinate  axes  are  rectangular,  is 


or 


(Y-y)^  +  X-x  =  o, 
v        y}  dx 

du  /T^       v       du  . 
—■(Y-ij)  =—  (X-x). 
dxK        JJ      dy  K  J 


(3) 


The  points  at  which  normals  are  parallel  to  the  line 
y  =  mx  +  n  are  given  by  aid  of  the  equation  of  the  curve  u  =  o 
along  with  the  equation 

du  du 

dy  dx' 

Examples. 


1.  Find  the  equation  of  the  normal  at  any  point  (x,  y)  on  the  ellipse 

/2 


x"      y 
a2      ¥ 


A       a*X      FY 

Am, =  a1  -  P. 

x  y 


2.  Find  the  equation  of  the  normal  at  any  point  on  the  curve 

ym  —  axn.  j_m.  nYy  +  mXx  =  ny2  -f  mx1. 

171.  Subtangent  and  Subnormal. — In  the  accom- 
panying figure,  let  PT  repre-  y 
sent  the  tangent  at  the  point  P, 
PiV the  normal;  OM,  PM the 
co-ordinates  at  P ;  then  the 
lines  TM  and  MJST  are  called 
the  subtangent  and  subnormal 
corresponding  to  the  point  P.  Fig.  9. 

To  find  the  expressions  for  their  lengths,  let  <p  =  L  PTM, 


then 


dx 


MN     ,  Ay 


UN=y% 

dx 


2 1 6  Tangents  and  Normals  to  Curves, 

The  lengths  of  FT  and  FN  are  sometimes  called  the 
lengths  of  the  tangent  and  the  normal  at  F :  it  is  easily- 
seen  that 


■JfJI  +  (sJ 


jj1+(tj 


PN=yJi  +  [^-\,      FT 

dp 

dx 


Examples. 

i.  To  find  the  length  of  the  subnormal  in  the  ellipse 

x2      y2 
~a2+¥=zl' 

Here  y—  =  -~x; 

ax         a* 

the  negative  sign  signifies  that  MN  is  measured  from  M  in  the  negative 
direction  along  the  axis  of  x,  i.e.  the  point  iVlies  between  M  and  the  centre  0 ; 
as  is  also  evident  from  the  shape  of  the  curve. 

2.  Prove  that  the  subtangent  in  the  logarithmic  curve,  y  —  ax,  is  of  constant 
length. 

3.  Prove  that  the  subnormal  in  the  parabola,  y2  =  zmcc,  is  equal  to  m. 

4.  Find  the  length  of  the  part  of  the  normal  to  the  catenary 


i{^+e~~°)> 


a  1 

y 


yi 

intercepted  by  the  axis  of  #.  Ans.  — . 

a 

5.  Find  at  what  point  the  subtangent  to  the  curve  whose  equation  is 

xy2  =  a2  (a  —  x) 

is  a  maximum.  Ans.  x  =  -,     y  —  a. 

172.  Perpendicular  on  Tangent. — Let  p  be  the  length 
of  the  perpendicular  from  the  origin  on  the  tangent  at  any 
point  on  the  curve 

F{x,  y)  =  c, 


Length  of  Perpendicular  on  Tangent.  217 

then  the  equation  of  the  tangent  may  be  written 

X  cos  w  +  T  sin  o>  =  p, 

where  w  is  the  angle  which  the  perpendicular  makes  with 
the  axis  of  x. 

Denoting  F  (x,  y)  by  u,  and  comparing  this  form  of  the 
equation  with  that  in  (2),  and  representing  the  common  value 
of  the  fraction  by  A, 

du  du  du        du 

—        — -      x—  +  y  ~r 
,  dx  dy         dx        ay  _  . 

°  cos  (v     sin  d)  p 

»-(£MD' 

du        du 
x—  +y  — 
-,  dx        dy  t  \ 

and  p=  J      •  (4) 

4\dx)  +\dk/J 

Cor.  If  .F(#,  y)  be  a  homogeneous  expression  of  the  nth 
degree  in  x  andy,  then  by  Euler's  formula,  Art.  102,  we  have 

du        du 
x—  +  y  —  =  nu  =  nc, 
dx        dy 

and  the  expression  for  the  length   of  the    perpendicular 
becomes  in  this  case 

nc 


} 


173.  In  the  curve 


duV    fduY 
dx)     \dy) 


xm     ym 


to  prove  that 

m  m  m 

p™11  =  (a  cos  w)J^r  +  (b  sin  a.)*"1"1.  (5) 


2 1 8  Tangents  and  Normals  to  Curves. 

By  Ex.  2,  Art.  169,  the  equation  of  the  tangent  is 

Xxm'1      Yym~l 

comparing  this  with  the  form 

X  cos  id  +  Y  sin  10  =  p9 


we  get 


or 


cos  u>     xm~l  sin  w     ym~x 

p     ~~a™'  p    =1ri 


a  cos  wY^1   x        (b  sin  oAm-1    y 


p      J        a'        \     p     J        b' 


Hence,  substituting  in  the  equation  of  the  curve,  we  obtain 
the  result  required. 

174.  l<ocus  of  Foot  of  Perpendicular  for  the  same 
Curve. — Let  X,  Yhe  the  co-ordinates  of  the  point  in  ques- 

X  Y 

tion,  and  we  have,  evidently,  cos  w  =  — ,  sin  o>  =  — :  substi- 
tuting these  values  for  cos  o>  and  sin  w  in  (5),  it  becomes 


(X2  +  Y2)m'1  =  {aX)m~l  +  (b  Y) 


m 
m-i 


since  p2  =  X2  +  Y2. 

175.  Another  Form  of  the  Equation  to  a  Tan- 
gent.— If  the  equation  of  a  curve  of  the  nth  degree  be 
written  in  the  form 

<p(x,  y)  =  Un  +  Un^  +  Un_2  +  .  .  .  +  U2  +  Ui  +  Un  =  o, 

where  un  denotes  the  homogeneous  part  of  the  nth  degree  in 
the  equation,  un^  that  of  the  [n  -  i)th,  &c. ;  then,  by  Cor. 
Art.  103,  we  have 

x    7     +  Vi      =  ~  lUn-i  +  2Wn_2  +  &C  .  .  .  +  nU0) . 

ax        ay 


Number  of  Tangents  from  an  External  Point.         219 

Hence  the  equation  of  the  tangent  in  Art.  169  becomes 

X-p  +  Y  -r-  +  un^  +  2Un_i  +  .  .  .  +  nii0  =  o  ;  (6) 

ax  ay 

an  equation  of  the  (n  -  i)th  degree  in  x  and  y. 

176.  Number  of  Tangents  from  an  External 
Point. — To  find  the  number  of  tangents  which  can  be 
drawn  to  a  curve  of  the  nth  degree  from  a  point  (a,  j3),  we  sub- 
stitute a  for  X,  and  [5  for  Y  in  (6) ,  and  it  becomes 

dd>        d$  ,  A 

a  -j-  +  p—  +  Un^i  +  2W«_a  +  .  .  .  +  flU0  =  O.  (7) 

dy»i/        ay 

This  represents  a  curve  of  the  (n  -  i)th  degree  in  x  and  y, 
and  the  points  of  its  intersection  with  the  given  curve  are  the 
points  of  contact  of  all  the  tangents  which  can  be  drawn 
from  the  point  (a,  (5)  to  the  curve.  Moreover,  as  two  curves 
of  the  degrees  n  and  n  -  \  intersect  in  general  in  n  (n  -  i) 
points,  real  or  imaginary  (Salmon's  Conic  Sections,  Art.  214), 
it  follows  that  there  can  in  general  be  n{n  -  1)  real  or 
imaginary  tangents  drawn  from  an  external  point  to  a  curve 
of  the  nth  degree. 

If  the  curve  be  of  the  second  degree,  equation  (7)  be- 
comes 

d(b       r^dcb 
a-!-+l5-j--  +  Ul+  2U0  =  O, 

ax        ay 

an  equation  of  the  first  degree,  which  evidently  represents 
the  polar  of  (a,  |3)  with  respect  to  the  conic. 
In  the  curve  of  the  third  degree 

uz  +  u2  +  ux  +  u0  =  o, 

equation  (7)  becomes 

d(h  d6 

which  represents  a  conic  that  passes  through  the  points  of 
contact  of  the  tangents  to  the  curve  from  the  point  (a,  |3). 

This  conic  is  called  the  polar  conic  of  the  point.  For  the 
origin  it  becomes 

U%  +  2th  +  3^0  =  o. 


220  Tangents  and  Normals  to  Curves. 

177.  Number  of  Normals  which  pass  through  a 
Given  Point. — If  a  normal  pass  through  the  point  (a,  j3), 
we  must  have  from  (3), 

.         x  an      .  ~.       x  ate 
(„-*)_=  (0-j,)-. 

This  represents  a  curve  of  the  nth  degree,  which  intersects  the 
given  curve  in  general  in  n2  points,  real  or  imaginary,  the 
normals  at  which  all  pass  through  the  point  (a,  j3). 
For  example,  the  points  on  the  ellipse 

x2     y2 

a~*  +  ¥=I' 

at  which  the  normals  pass  through  a  given  point  (a,  j3), 
are  determined  by  the  intersection  of  the  ellipse  with  the 
hyperbola 

xy(a2  -  b2)  =  a2  ay  -  b2(5x. 

For  the  modification  in  the  results  of  this  and  the  pre- 
ceding article  arising  from  the  existence  of  singular  points  on 
the  curve,  the  student  is  referred  to  Salmon's  Higher  Plane 
Curves,  Arts.  66,  67,  in. 

178.  Differential  of  the  Arc  of  a  Plane  Curve. 
Direction  of  the  Tangent. — If  the  length  of  the  arc  of  a 
curve,  measured  from  a  fixed  point  A  on  it,  be  denoted  by  s, 
then  an  infinitely  small  portion  of  it  is  represented  by  ds. 
Again,  if  $'  represent  the  angle  QPL  (fig.  8),  we  have 

,     PL       ,    .      ,       QL 
cos  <p  =  jg,  and  sm0   =  ^; 

but  in  the  limit,  PL  =  dx,  QL  =  dy,  and  PQ  =  ds*  and  also 
(j/  becomes  PTX,  or  <j>  (fig.  9). 


*  In  Art.  37  it  has  been  proved  that  the  difference  between  the  length  of  an 

infinitely  small  arc  and  its  chord  is  an  infinitely  small  quantity  of  the  second 

arc  PQ  -  PQ  . 
order  in  comparison  with  the  length  of  the  chord;  i.e. — is  infinitely 

small  of  the  second  order,  and  therefore  this  fraction  vanishes  in  the  limit. 

arc  PQ 

Hence  r-^  =  i>  ultimately. 

ord  PQ        '  J 


Differential  of  the  Arc  of  a  Curve. 


221 


Hence 


dx 


dy 


COS0=-,      on*      ^ 


squaring  and  adding,  we  get 

fdx\2  (dyY_ 
\dsj    \dsj 

Hence,  also,  we  have 

ds2  =  dx2  +  dy2, 


(8) 


(9) 


and  therefore 


ds 


(io) 


On  account  of  the  importance  of  these  results,  we  shall 
give  another  proof,  as  follows : — 

Let,  as  before,  PR  be  the  tangent  to  the  curve  at  the 
point  P, 

OM=x,  PM=y, 

MN=PL=Ax,  QL  =  Ay. 

Z.PTX=<j>,  arc  PQ=  As, 


Then,  if  the  curvature  of 
the  elementary  portion  PQ 
of  the  curve  be  continuous, 
we  have  evidently  the  line 

PQ<2ltgPQ<PB+QE;      o      t  m    n 

Fig.  io. 

or         ^  Ax%  +  Ay1  <  As  <  Ax  sec  0  +  Ay  -  Ax  tan  $ ; 


x 


j 


[AyV     As  Ay     . 

1  +  \-r-\  <-r-  <  sec  <b  +  -r-  -  tand. 
\Ax)      Ax  r     Ax  r 


Again,  in  the  limit  — -  =  -~  =  tan  6,  and  Ji  +  (  — ) 

Ax     dx  r  V        \AxJ 


222 


Tangents  and  Normals  to  Curves. 


dy 


=  Ji  + 


dy 

dx 


■which   establishes  the  required 


becomes  J  i  +  I  -j- 1  or  sec  0  ;  accordingly  each  of  the  pre- 
ceding expressions  converges  to  the  same  limiting  value,  and 

,         ds        I 
we  nave  -7-  =  A  /  ] 
dx     \ 

result. 

179.  Polar  Co-ordinates. — The  position  of  any  point 
in  a  plane  is  determined  when  its  distance  from  a  fixed  point 
called  a  pole,  and  the  angle  which  that  distance  makes  with  a 
fixed  line,  are  known  ;  these  are  called  the  polar  co-ordinates 
of  the  point,  and  are  usually  denoted  by  the  letters  r  and  0. 
The  fixed  line  is  called  the  prime  vector,  and  r  is  called  the 
radius  vector  of  the  point. 

The  equation  of  a  curve  referred  to  polar  co-ordinates  is 
generally  written  in  one  or  other  of  the  forms, 

r  =/(0),  or  F(r,  0)  =  o, 

according  as  r  is  given  explicitly  or  implicitly  in  terms  of  6. 
Also,  if  0  be  positive  when  measured  above  the  prime  vector, 
it  must  be  regarded  as  negative  when  measured  below  it. 

1 80.  Angle  between  Tangent  and  Radius  Vector. 
Let  0  be  the  pole,  P  and  Q  two  near 
points  on  the  curve,  PM  a  perpendicular 
on  OQ,  OP  =  r,  POX  =  0,  and  if,  the 
angle  between  the  tangent  and  radius 
vector.     Then 


tan  OQP  = 


PM 


QM' 


sin  OQP  = 


PM 
PQ' 


cos  OQP  = 


QM 


,  but  in  the  limit  when 

Q  and  P  coincide,    the   angle    OQP 
becomes  equal  to  \p,  and* 


Fig. 


11. 


QM  _  dr 

PQ~  ds' 


-p7^  =  — ,  at  the  same  time ; 


or 


dr        „  rdO 

C0Sxfj  =  ds~'     Sm^=~ds~'     tan^  = 


rdO 
dr' 


(") 


*  These  results  can  be  easily  established  from  Art.  37. 


Polar  Subtangent  and  Subnormal. 


223 


Also, 


rdO 

ds 


dr 


-(-7-1=1. 
Ms 


(12) 


Hence,   also,   we  can  determine   an   expression  for  the 
differential  of  an  arc  in  polar  co-ordinates ;  for,  since 

PQ2  PMZ 

QM2  ~  I  +  QM0'' 

we  get,  on  proceeding  to  the  limit, 

ds_       J        r2dQ2 
dr      \  dr*  ' 


or 


ds  =      1  + 


r2d62 
dr% 


dr. 


(13) 


These  results  are  of  importance  in  the  general  theory  of 
curves. 

181.  Application    to    the    logarithmic    Spiral. — 

The  curve  whose  equation  is  r  =  a9  is  called  the  logarithmic 
spiral.     In  this  curve  we  have 

rdO  1 

tan  w  =  -7—  =  = . 

dr       log  a 

Accordingly,  the  angle  between  the  radius  vector  and  the 
tangent  is  constant.  On  account  of  this  property  the  curve 
is  also  called  the  equiangular  spiral. 

182.  Polar  Subtangent  and  Subnormal. — Through 
the  origin  0  let  ST  be  drawn  perpendi-     g 

cular  to  OP,  meeting  the  tangent  in  T, 

and  the  normal  in  S.   The  lines  0  T  and 

OS  are  called  the  polar  subtangent  and 

subnormal,  for  the  point  P.     To  find 

their  values,  we  have 

r*dO 
OT  =  OP  tan  OPT  =  r  tan  $  =  ~. 

cir 


OS  =  OP  tan  OPS  =  r  cot  1/,  = 


Also,  if 


1 

u  =  -, 

r 


dr  "") 

dff  I 

OT=-f.  \ 

du  J 


224  Tangents  and  Normals  to  Curves. 

Again,  if  ON  he  drawn  perpendicular  to  PT,  we  hav© 

dr 
PN  =  OP  eosxp  =  r—.  (15) 

as 

183.  Expression  for  Perpendicular  on  Tangent. — 

As  before,  let  p  =  ON,  then 

.     ,      r2dO 

p  =  r  sin  \p  = 


hence 


ds    ' 


ds*       drz  +  r2d62       dr2        1 

+ 


p*     r*d&  rW  r*dd2     r2' 


1        „      fdu\*  ,  ^ 

?=M+UJ-  (l6) 

The  equations  in  polar  co-ordinates  of  the  tangent  and 
the  normal  at  any  point  on  a  curve  can  be  found  without 
difficulty :  they  have,  however,  been  omitted  here,  as  they 
are  of  little  or  no  practical  advantage. 


Examples. 

1.  To  find  the  length  of  the  perpendicular  from  a  focus  on  the  tangent  to  an 
ellipse. 

The  focal  equation  of  the  curve  is 

a(i  -  e2)  1  —  e  cos  0 

r  = '-,  or  u=  — -r- ; 

I  -  e  cos  0  a  (i  -  e2) 

.  du        e  sin  0 

hence  —  = : 

dd      a(i-e*)' 

I  I  +02  -  20  COS0  _  I  /2U  \ 

*V=      a2(i-e2)2  «»(i  -e2)  \~r  ~     / 

2.  Prove  that  the  polar  subnormal  is  constant,  in  the  curve  r  *=  ad ;  and  the 
nolar  subtangent,  in  the  curvo  rfl  =  a. 


Inverse  Curves.  225 

184.  Inverse  Curves. — If  on  any  radius  vector  OP, 
drawn  from  a  fixed  origin  0,  a  point  P'  be  taken  such  that 
the  rectangle  OP  .  OP'  is  constant,  the  point  P'  is  called  the 
inverse  of  the  point  P ;  and  if  P  describe  any  curve,  Pf 
describes  another  curve  called  the  inverse  of  the  former. 

The  polar  equation  of  the  inverse  is  obtained  immediately 
from  that  of  the  original  curve  by 

substituting  —  instead  of  r  in  its 

equation ;  where  W  is  equal  to  the 
constant  OP  .  OP'. 

Again,  let  P,  Q  be  two  points, 
and  P',  Q'  the  inverse  points  ;  then 
since  OP  .  OP'  =  OQ  .  OQ',  the 
four  points  P,  Q,  Q',  P',  lie  on  a 
circle,  and  hence  the  triangles 
OQP  and  OP'Q'  are  equiangular  ; 

PQ       OP      OP  .  OQ      OP.  OQ 
'''  P'Q'~  0Q'~  OQ.OQ'~       ¥       '  (I7> 

Again,  if  P,  Q  be  infinitely  near  points,  denoting  the 
lengths  of  the  corresponding  elements  of  the  curve  and  of  its 
inverse  by  ds  and  ds',  the  preceding  result  becomes 


ds=T-ds'.  (18) 


185.  Direction  of  the  Tangent  to  an  Inverse 
Curve. — Let  the  points  P,  Q  belong  to  one  curve,  and  P',  Q' 
to  its  inverse ;  then  when  P  and  Q  coincide,  the  lines  PQ, 
P'Q'  become  the  tangents  at  the  inverse  points  P  and  P' : 
again,  since  the  angle  SPP'  =  the  angle  SQ'Q,  it  follows  that 
the  tangents  at  P  and  P'  form  an  isosceles  triangle  with  the 
line  PP'. 

By  aid  of  this  property  the  tangent  at  any  point  on  a 
curve  can  be  drawn,  whenever  that  at  the  corresponding 
point  of  the  inverse  curve  is  known. 

It  follows  immediately  from  the  preceding  result,  that  if 
two  curves  intersect  at  any  angle,  their  inverse  curves  intersect  at 
the  same  angle. 

Q 


226  Tangents  and  Normals  to  Curves. 

1 86.  .Equation  to  the  Inverse  of  a  Given  Curve. — 

Suppose  the  curve  referred  to  rectangular  axes  drawn  through, 
the  pole  0,  and  that  as  and  y  are  the  co-ordinates  of  a  point  P 
on  the  curve,  X  and  Y  those  of  the  inverse  point  Pf ;  then 

x_  _  op_  _  op  .  or  _      k2       .  y_        k2 

X~  0F~      OF2      "  X2  +  Y2'  similari^  y"  X2  +  Y2 ; 

hence  the  equation  of  the  inverse  is  got  by  substituting 

k2x          ,     k2y 
and *—z 


x2  +  y2         x*  +  y 

instead  of  x  and  y  in  the  equation  of  the  original  curve 

Again,  let  the  equation  of  the  original  curve,  as  in  Art. 
174,  be 

Un  +  Un-i  +  un_2  +  .  .  .  +  u2  +  ux  +  u0  =  o. 

When  — and  -r— — -  are  substituted  for  x  and  y.  un 

x2  +  y2         x2  +  y2 

k2nu 
becomes  evidently  —^ — ^— . 
J   (x2  +  y2)n 

Accordingly,  the  equation  of  the  inverse  curve  is 
k2nun  +  k2n-2u.n_x  (x2  +  y2)  +  W^u^x2  +  y2)2  +  .  .  . 

+  u0  (x2  +  y2)n  =  o.  (19) 

For  instance,  the  equation  of  any  right  line  is  of  the  form 

uL  +  u0  =  o ; 
hence  that  of  its  inverse  with  respect  to  the  origin  is 
Wux  +  u0  (x2  +  y2)  =  o. 

This  represents  a  circle  passing  through  the  pole,  as  is 
well  known,  except  when  u0  =  o ;  i.e.  when  the  line  passes 
through  the  pole  0. 

Again,  the  equation  of  the  inverse  of  the  circle 

x2  +  y2  +  ux  +  u0  =  o, 

with  respect  to  the  origin,  is 

(&4  +  k2Ui  +  u0(x2  +  y2))  (x2  +  y2)  =  o, 

which  represents  another  circle^  along  with  the  two  imaginary 
right  lines  x2  +  y2  =  o. 


Pedal  Curves. 


227 


Again,  the  general  equation  of  a  conic  is  of  the  form 

%h  +  ux  +  u0  =  o ; 

hence  that  of  its  inverse  with  respect  to  the  origin  is 

&%2  +  k2ih(%2  +  y2)  +  w0(#2  +  y2)2  =  o, 

which  represents  a  curve  of  the  fourth  degree  of  the  class 
called  "bicircular  quartics." 

If  the  origin  be  on  the  conic  the  absolute  term  u0  vanishes, 
and  the  inverse  is  the  curve  of  the  third  degree  represented 

by 

k2u2  +  Ui  (x2  +  y2)  =  o. 

This  curve  is  called  a  "  circular  cubic." 

If  the  focus  be  the  origin  of  inversion,  the  inverse  is  a 
curve  called  the  Limacon  of  Pascal.  The  form  of  this  curve 
will  be  given  in  a  subsequent  Chapter. 

187.  Pedal  Curves. — If  from  any  point  as  origin  a  per- 
pendicular be  drawn  to  the  tangent  to  a  given  curve,  the  locus 
of  the  foot  of  the  perpendicular  is  called  the  pedal  of  the  curve 
with  respect  to  the  assumed  origin. 

In  like  manner,  if  perpendiculars  be  drawn  to  the  tan- 
gents to  the  pedal,  we  get  a  new  curve  called  the  second  pedal 
of  the  original,  and  so  on.  With  respect  to  its  pedal,  the 
original  curve  is  styled  the  first  negative  pedal,  &c. 

188.  Tangent  at  any  Point  to  the  Pedal  of  a 
given  Curve. — Let  ON,  ON' 
be  the  perpendiculars  from  the 
origin  0  on  the  tangents  drawn 
at  two  points  P  and  Q  on  the 
given  curve,  and  J1  the  intersec- 
tion of  these  tangents ;  join  NN'; 
then  since  the  angles  ONT  and 
ONfT  are  right  angles,  the  qua- 
drilateral ON  N'T  is  inscribable 
in  a  circle, 

.-.  lONN=lOTN 

In  the  limit  when  P  and  Q  coincide,  L  OTN  =  L  OPN, 
and  NN'  becomes  the  tangent  to  the  locus  of  N;  hence  the 

q  2 


228  Tangents  and  Normals  to  Curves. 

latter  tangent  makes  the   same  angle  with  ON  that  the 

tangent  at  P  makes  with  OP.     This  property  enables  us 

to  draw  the  tangent  at  any  point  N  on  the  pedal  locus  in 

question. 

Again,  if  p'  represent  the  perpendicular  on  the  tangent  at 

N  to  the  first  pedal,  from  similar  triangles  we  evidently  have 

p2 
r  =  —. 

p 

Hence,  if  the  equation  of  a  curve  he  given  in  the  form 

p2 
r  =f{p),  that  of  its  first  pedal  is  of  the  form  —  =f(p),  in 

which  p  and  pf  are  respectively  analogous  to  r  and  p  in  the 
original  curve.  In  like  manner  the  equation  of  the  next 
pedal  can  be  determined,  and  so  on. 

189.  Reciprocal  Polars. — If  on  the  perpendicular  ON 
a  point  P'  be  taken,  such  that  OP'.  ON  is  constant  (k2  sup- 
pose), the  point  P'  is  evidently  the  pole  of  the  line  PN  with 
respect  to  the  circle  of  radius  k  and  centre  0 ;  and  if  all  the 
tangents  to  the  curve  be  taken,  the  locus  of  their  poles  is  a 
new  curve.  We  shall  denote  these  curves  by  the  letters  A 
and  P,  respectively.  Again,  by  elementary  geometry,  the 
point  of  intersection  of  any  two  lines  is  the  pole  of  the  line 
joining  the  poles  of  the  lines*  Now,  if  the  lines  be  taken  as 
two  infinitely  near  tangents  to  the  curve  A,  the  line  joining 
their  poles  becomes  a  tangent  to  B ;  accordingly,  the  tangent 
to  the  curve  B  has  its  pole  on  the  curve  A.  Hence  A  is  the 
locus  of  the  poles  of  the  tangents  to  B. 

In  consequence  of  this  reciprocal  relation,  the  curves  A  and 
B  are  called  reciprocal  polar s  of  each  other  with  respect  to  the 
circle  whose  radius  is  k. 

Since  to  every  tangent  to  a  curve  corresponds  a  point  on 
its  reciprocal  polar,  it  follows  that  to  a  number  of  points  in 
directum  on  one  curve  correspond  a  number  of  tangents  to  its 
reciprocal  polar,  which  pass  through  a  common  point. 

Again,  it  is  evident  that  the  reciprocal  polar  to  any  curve 
is  the  inverse  to  its  pedal  with  respect  to  the  origin. 

"We  have  seen  in  Art.  1 80  that  the  greatest  number  of  tan- 
gents from  a  point  to  a  curve  of  the  nth  degree  is  n(n  -  1)  ; 

*  Townsend's  Modern  Geometry,  vol.  i.,  p.  219. 


Reciprocal  Polars.  229 

hence  the  greatest  number  of  points  in  which  its  reciprocal 
polar  can  be  cut  by  a  line  is  n(n  —  1),  or  the  degree  of  the 
reciprocal  polar  is  n  (n-  1 ) .  For  the  modification  in  this 
result,  arising  from  singular  points  in  the  original  curve,  as 
well  as  for  the  complete  discussion  of  reciprocal  polars,  the 
student  is  referred  to  Salmon's  Higher  Plane  Curves. 

As  an  example  of  reciprocal  polars  we  shall  take  the  curve 
considered  in  Art.  173. 

If  r  denote  the  radius  vector  of  the  reciprocal  polar  cor- 
responding to  the  perpendicular  p  in  the  proposed  curve,  we 
have 

¥ 

Substituting  this  value  for^>  in  equation  (5),  we  get 

m  m  m 

fk~\m~i  m-l  .  m-l 

i—j     =  (a  cos  to)     +  (0  sin  to)      , 

2m  m  m 

or  I™1*  =  (axY~x  +  {byf^\ 

which  is  the  equation  of  the  reciprocal  polar  of  the  curve  re- 
presented by  the  equation 

—  +  7T-  =   I. 

am     bm 

In  the  particular  case  of  the  ellipse, 

x2     if 

a%      b%        ' 

the  reciprocal  polar  has  for  its  equation 

¥  =  aV  +  5y. 

The  theory  of  reciprocal  polars  indicated  above  admits  of 
easy  generalization.  Thus,  if  we  take  the  poles  with  respect 
to  any  conic  section  ( U)  of  all  the  tangents  to  a  given  curve 
A,  we  shall  get  a  new  curve  B ;  and  it  can  be  easily  seen,  as 
before,  that  the  poles  of  the  tangents  to  B  are  situated  on  the 
curve  A.  Hence  the  curves  are  said  to  be  reciprocal  polars 
with  respect  to  the  conic  TJ. 

It  may  be  added,  that  if  two  curves  have  a  common  point, 


230  Tangents  and  Normals  to  Curves. 

their  reciprocal  polars  have  a  common  tangent;  and  if  the 
curves  touch,  their  reciprocal  polars  also  touch. 

For  illustrations  of  the  great  importance  of  this  "  principle 
of  duality,"  and  of  reciprocal  polars  as  a  method  of  investi- 
gation, the  student  is  referred  to  Salmon's  Conies,  ch.  xv. 

We  next  proceed  to  illustrate  the  preceding  by  discussing 
a  few  elementary  properties  of  the  curves  which  are  comprised 
under  the  equation  rm  =  am  cos  mO. 

190.  Pedal  and  Reciprocal  Polar  ofrm  =  am  cos  mO. 
We  shall  commence   by  finding  the  ^ 

angle  between  the  radius  vector  and 
the  perpendicular  on  the  tangent. 

In  the   accompanying  figure   we 

have  tan  PON  =  cot  OPN  =-■%,.         /-^ 

rdd         0 

Fig.  15- 
But  m  log  r  =  m  log  a  +  log  (cos  mO) ; 

dr 
hence  —Q  =  -  tan  mO, 

rclu 

and  accordingly,  LPON=mO.  (20) 


7* 


WJ+1 


Again,  p  =  ON  =  r  eosmO  = 

or  rm+1  =  amp.  (21) 

The  equation  of  the  pedal,  with  respect  to  0,  can  be  im- 
mediately found. 

For,  let  l  A  ON  =  w,  and  we  have 

a)  =  (m  +1)6. 


Also,  from  (21),         [-)-[-)     ■ 
Hence,  the  equation  of  the  pedal  is 


■  »»'+i  =  am+1  cos    .  {22) 

\m  +  1  ' 


On  the  Curve  rm  =  am  cos  mQ.  231 

Consequently,  the  equation  of  the  pedal  is  got  by  substi- 

tuting  instead  of  m  in  the  equation  of  the  curve. 

°  m  +  1 

By  a  like  substitution  the  equation  of  the  second  pedal  is 
easily  seen  to  be 

mm  f\ 

—        • —  mu 

™2m+i  _  /,2m+i 


=  a""^1  cos ; 

2m  +  1 

and  that  of  the  nth  pedal 

m           J>L_             mQ 
rmn+l  =  amn+\  cog p  /2  ,\ 

WW   +   1 

Again,  from  Art.  1 84,  it  is  plain  that  the  inverse  to  the 
curve  rm  =  am  cos  md,  with  respect  to  a  circle  of  radius  a,  is 
the  curve  rm  cos  mO  =  am. 

Again,  the  reciprocal  polar  of  the  proposed,  with  respect 
to  the  same  circle,  being  the  inverse  of  its  pedal,  is  the  curve 

m  n  m 

—  mu  — ■ 

rro+i   cog =    am+K  f2.\ 

m+i 
It  may  be  observed  that  this  equation  is  got  by  substitut- 

/yyi 

ins:  for  m  in  the  original  equation. 

0  m  +  1 

Accordingly  we  see  that  the  pedals,  inverse  curves,  and 
reciprocal  polars  of  the  proposed,  are  all  curves  whose  equa- 
tions are  of  the  same  form  as  that  of  the  proposed. 

In  a  subsequent  chapter  the  student  will  find  an  additional 
discussion  of  this  class  of  curves,  along  with  illustrations  of 
their  shape  for  a  few  particular  values  of  m. 

Examples. 

1 .  The  equation  of  a  parabola  referred  to  its  focus  as  pole  is 

r  (1  +  cos  6)  =  2a, 
to  find  the  relation  between  r  and  p. 

a 

Here  H  cos  -  =  ah,  and  consequently  p2  =  ar, 

a  well-known  elementary  property  of  the  curve. 


232  Tangents  and  Normals  to  Curves. 

2.  The  equation  r2  cos  20  =  a2  represents  an  equilateral  hyperbola ;  prove 
that^r  =  a2. 

3.  Trie  equation  r2  =  a2  cos  2O  represents  a  Lemniscate  of  Bernoulli ;  find 
the  equation  connecting  p  and  r  in  this  case.  Ans.  r3  —  a2p. 

4.  Find  the  equation  connecting  the  radius  vector  and  the  perpendicular  on 
the  tangent  in  the  Cardioid  whose  equation  is 

r  =  a(i  +  cos  9).  Ans.  r3  =  2ap2. 

It  is  evident  that  the  Cardioid  is  the  inverse  of  a  parabola  with  respect  to 
its  focus  ;  and  the  Lemniscate  that  of  an  equilateral  hyperbola  with  respect  to 
its  centre.  Accordingly,  we  can  easily  draw  the  tangents  at  any  point  on  either 
of  these  curves  by  aid  of  the  Theorem  of  Art.  185. 

5.  Show,  by  the  method  of  Art.  188,  that  the  pedal  of  the  parabola,  p2  =  ar, 
with  respect  to  its  focus,  is  the  right  line  p  =  a. 

6.  Show  that  the  pedal  of  the  equilateral  hyperbola  pr  =  a1  is  a  Lemniscate. 

7.  Find  the  pedal  of  the  circle  r2  =  iap.         Ans.  A  Cardioid,  r3  =  2ap2. 

191.  Expression  for  PN — To  find  the  value  of  the 
intercept  between  the  point  of 
contact  P'and  the  foot  N  of 
the    perpendicular    from    the 
origin  on  the  tangent  at  P. 

Let  p=ON,u>=L  NO  A,  ^ 
PN=t;  then  z_NTN'=LNON' 
=  Aw,  also  SN'=TS  sinSTN; 
SN' 

sin  NON 


TS  =    :    ^^r/ ;  hut  in  the 


CLYi 

limit,  when  PQ  is  infinitely  small,  -: — ^ftttf/  becomes  -=-, 
and  TS  becomes  PN  or  t ; 


Also  OP2  =  ON2  +  PN2 


,^f,  [fj.  (>6) 


192.  To  prove  that 


ds  dt  /     v 

-r=p+T.  (27) 


Vectorial  Co-ordinates.  233 

On  reference  to  the  last  figure  we  have 

ds      ..    .,    £PT+TQ      dt      ..    .,    £QN'-PN 

—  =  limit  of  ,     —  =  limit  01 ; 

da)  Aw  dd>  Aw 

but  PT  +  TQ  -  QNf  +  PN  =  TJST  -  TN', 

,  ds      dt     ,.    .,     „TN-TN'     v    .,    ,SN    niKT 

hence —  =  limit  01 =  limit  01  - —  =  OJy=p ; 

dhi      do)  Aw  Aw 

ds  dt 

da)  db) 

This  result,  which  is  due  to  Legendre,  is  of  importance  in 
the  Integral  Calculus,  in  connexion  with  the  rectification  of 
curves. 

If  -~-  be  substituted  for  t,  the  preceding  formula  becomes 
doj 

dio  dojz 

This  shape  of  the  result  is  of  use  in  connexion  with  curva- 
ture, as  will  be  seen  in  a  subsequent  chapter. 

193.  Direction  of  lormal  in  Tectorial  Co-ordi- 
nates.— In  some  cases  the  equation  of  a  curve  can  be 
expressed  in  terms  of  the  distances  from  two  or  more  fixed 
points  or  foci.  Such  distances  are  called  vectorial  co-ordi- 
nates. For  instance,  if  rl9  r%  denote  the  distances  from  two 
fixed  points,  the  equation  rx  +  r2  -  const,  represents  an  ellipse, 
and  rx  —  r%  -  const.,  a  hyperbola. 

Again,  the  equation 

ri  +  mr-z  =  const. 

represents  a  curve  called  a  Cartesian*  oval. 
Also,  the  equation 

7*1  r2  =  const. 

represents  an  oval  of  Cassini,  and  so  on. 

The  direction  of  the  normal  at  any  point  of  a  curve,  in 
such  cases,  can  be  readily  obtained  by  a  geometrical  con- 
struction. 

*  A  discussion  of  the  principal  properties  of  Cartesian  ovals  will  be  found 
in  Chapter  XX. 


234  Tangents  and  Normals  to  Curves. 

For,  let 

F(n,  r2)  =  const. 
be  the  equation  of  the  curve,  where 

F1P  =  r1,     FzP  =  r2, 
then  we  have 

dFdrv      dFdr2 
di\  ds      dr%  ds 

Now,  if  PThe  the  tangent  at  P,  then,  by  Art.  1 80,  we  have 

dr  dv 

—  =  cos  ^1,    -^  =  cos  i/>2,     where  ^  =  l  TPFl9     ?//2  =  z.  TPF2. 

TT  dF  .         dF  .  ' 

xtence  —  cos  \pi+  —  cos  1/- 2  =  o.  (29) 

ctr-i  Qj)"% 

Again,  from  any  point  R  on  the  normal  draw  RL  and 
R M  respectively  parallel  to  F2P  and  FXP9  and  we  have 

PL  :  LR  =  sin  RPM :  sin  RPL  =  cos  ^2  :  -  cos  & 

_dF  m  dF 
drx  '  dr2' 
Accordingly,   if  we  measure  on  PFi  and  PF%  lengths 

PL  and  PM9  which  are  in  the  proportion  of  —  to  — ,  then 

ar\       ar^ 

the  diagonal  of  the  parallelogram  thus  formed  is  the  normal 

required. 

This  result  admits  of  the  following  generalization  : 
Let  the  equation  of  the  curve*  be  represented  by 

F{n,  rt9  n,  •  •  .  rn)  =  const., 

*  The  theorem  given  above  is  taken  from  Poinsot's  Elements  de  Statique, 
Neuvieme  Edition,  p.  435.  The  principle  on  which  it  was  founded  was,  how- 
ever, given  by  Leibnitz  (Journal  des  Savans,  1693),  and  was  deduced  from 
mechanical  considerations.  The  term  resultant  is  borrowed  from  Mechanics, 
and  is  obtained  by  the  same  construction  as  that  for  the  resultant  of  a  number 
of  forces  acting  at  the  same  point.  Thus,  to  find  the  resultant  of  a  number  of 
lines  Fa,  Pb,  Pc,  Pd,  .  .  .  issuing  from  a  point  P,  we  draw  through  a  a  right 
line  aB,  equal  and  parallel  to  Pb,  and  in  the  same  direction ;  through  B,  a  right 
line  BC,  equal  and  parallel  to  Pc,  and  so  on,  whatever  be  the  number  of  lines: 
then  the  bine  US,  which  closes  the  polygon,  is  the  resultant  in  question. 


Normals  in  Vectorial  Co-ordinates.  235 

where  riy  r2,  .  .  .  rn  denote  the  distances  from  n  fixed  points. 
To  draw  the  normal  at  any  point,  we  connect  the  point  with 
the  n  fixed  points,  and  on  the  joining  lines  measure  off 
lengths  proportional  to 

dF     dF     dF  dF  ,.    , 

^-'     J"'     -J->  •  •  •  7">  respectively; 
ai'i      dr-i      drz  clrn 

then  the  direction  of  the  normal  is  the  resultant  of  the  lines 
thus  determined. 

For,  as  before,  we  have 

dFdrx      dFdr2  dF  drn  _ 

drx  ds      dr2  ds  '  drn  ds 

it  dF        .      dF        .  dF  ' 

Hence       —  cos  fa  +  —  cos  fa  +  -  •  •  -7—  cos  ipw  =  o.  (30) 

drx  ar2  ar^ 

dF        ,      dF  dF 

JNOW,       — -COS^i,      —  COS  \pz,    ...     —  COS  \ln, 

arx  ar2  ar^i 

are  evidently  proportional  to  the  projections  on  the  tangent 
of  the  segments  measured  off  in  our  construction.  Moreover, 
in  any  polygon,  the  projection  of  one  side  on  any  right  line 
is  manifestly  equal  to  the  sum  of  the  projections  of  all  the 
other  sides  on  the  same  line,  taken  with  their  proper  signs. 
Consequently,  from  (30),  the  projection  of  the  resultant  on 
the  tangent  is  zero ;  and,  accordingly,  the  resultant  is  normal 
to  the  curve,  which  establishes  the  theorem. 

It  can  be  shown  without  difficulty  that  the  normal  at  any 
point  of  a  surface  whose  equation  is  given  in  terms  of  the 
distances  from  fixed  points  can  be  determined  by  the  same 
construction. 

Examples. 

1.  A  Cartesian  oval  is  the  locus  of  a  point,  P,  such  that  its  distances,  Pltl, 
PM\  from  the  circumferences  of  two  given  circles  are  to  each  other  in  a  constant 
ratio  ;  prove  geometrically  that  the  tangents  to  the  oval  at  P,  and  to  the  circles 
at  M  and  M ',  meet  in  the  same  point. 

2.  The  equation  of  an  ellipse  of  Cassini  is  rr'  =  ab,  where  r  and  /  are  the 
distances  of  any  point  P  on  the  curve,  from  two  fixed  points,  A  and  B.  If  0 
he  the  middle  point  of  AB,  and  PN the  normal  at  P,  prove  that  L  APO=  L  BPN. 

3.  In  the  curve  represented  hy  the  equation  rj.3  +  r2z  =  a3,  prove  that  the 
normal  divides  the  distance  hetween  the  foci  in  the  ratio  of  r2  to  n. 


236  Tangents  and  Normals  to  Curves. 

1 94.  In  like  manner,  if  the  equation  of  a  curve  be  given 
in  terms  of  the  angles  9U  02,  .  .  .  9n,  which  the  vectors  drawn 
to  fixed  points  make  respectively  with  a  fixed  right  line,  the 
direction  of  the  tangent  at  any  point  is  obtained  by  an  analo- 
gous construction. 

For,  let  the  equation  be  represented  by 

F(0l9  0a,  .  .  .  0„)  =  const. 

Then,  by  differentiation,  we  have 

dFdfa     dFa%  dF  dOn  _ 

dOi  ds      ddz  ds  '  ddn  ds 

Hence,  as  before,  from  Art.  180,  we  get 

1  dF  .     ,       1  dF  .     ,  1  dF   .     .  ,     N 

--sinf  +  --5-siinf2  +  .  .  -  +  —  JX-  sin  \frn  =  0.       (31) 
i\  ddx  r2  dVi  rn  ddn 

Accordingly,  if  we  measure  on  the  lines  drawn  to  the  fixed 
points  segments  proportional  to 

1  dF      1  dF  1  dF 

rx  dOi      r2  ddz   '     '  rn  ddn 

and  construct  the  resultant  line  as  before,  then  this  line  will 
be  the  tangent  required.  The  proof  is  identical  with  that  of 
last  Article. 

195.  Curves  Symmetrical  with  respect  to  a  l^ine, 
asad  Centres  of  Curves. — It  may  be  observed  here,  that 
if  the  equation  of  a  curve  be  unaltered  when  y  is  changed 
into  -  y,  then  to  every  value  of  x  correspond  equal  and  oppo- 
site values  of  y;  and,  when  the  co-ordinate  axes  are  rect- 
angular, the  curve  is  symmetrical  with  respect  to  the  axis  of  x. 

In  like  manner,  a  curve  is  symmetrical  with  respect  to 
the  axis  of  y,  if  its  equation  remains  unaltered  when  the  sign 
of  x  is  changed. 

Again,  if,  when  we  change  x  and  y  into  -  x  and  -  y,  re- 
spectively, the  equation  of  a  curve  remains  unaltered,  then 
every  right  line  drawn  through  the  origin  and  terminated  by 
the  curve  is  divided  into  equal  parts  at  the  origin.  This 
takes  place  for  a  curve  of  an  even  degree  when  the  sum  of 


Symmetrical  Curves  and  Centres.  237 

the  indices  of  x  and  y  in  each  term  is  even ;  and  for  a  curve 
of  an  odd  degree  when  the  like  sum  is  odd.  Such  a  point  is 
called  the  centre*  of  the  curve.  For  instance,  in  conies,  when 
the  equation  is  of  the  form 

ax2  +  2hxy  +  by2  =  c, 

the  origin  is  a  centre.  Also,  if  the  equation  of  a  cubicf  be 
reducible  to  the  form 

Uz  +  Ux  =  O, 

the  origin  is  a  centre,  and  every  line  drawn  through  it  is  bi- 
sected at  that  point. 

Thus  we  see  that  when  a  cubic  has  a  centre,  that  point 
lies  on  the  curve.  This  property  holds  for  all  curves  of  an 
odd  degree. 

It  should  be  observed  that  curves  of  higher  degrees  than 
the  second  cannot  generally  have  a  centre,  for  it  is  evidently 
impossible  by  transformation  of  co-ordinates  to  eliminate  the 
requisite  number  of  terms  from  the  equation  of  the  curve. 
For  instance,  to  seek  whether  a  cubic  has  a  centre,  we  substi- 
tute X  +  a  for  x,  and  T  +  j3  for  y,  in  its  equation,  and  equate 
to  zero  the  coefficients  of  X2,  XFand  Y2,  as  well  as  the  abso- 
lute term,  in  the  new  equation :  as  we  have  but  two  arbitrary 
constants  (a  and  j3)  to  satisfy  four  equations,  there  will  be 
two  equations  of  condition  among  its  constants  in  order  that 
the  cubic  should  have  a  centre.  The  number  of  conditions  is 
obviously  greater  for  curves  of  higher  degrees. 


*  For  a  general  meaning  of  the  word  "  centre,"  as  applied  to  curves  of 
higher  degrees,  see  Chasles's  Apercu  Historique,  p.  233,  note. 

f  This  name  has  heen  given  to  curves  of  the  third  degree  by  Dr.  Salmon, 
in  his  Higher  Plane  Curves,  and  has  heen  generally  adopted  hy  subsequent 
writers  on  the  subject. 


2  38  Examples. 


Examples. 

i  .  Find  the  lengths  of  the  subtangent  and  subnormal  at  any  point  of  the 
curve 


yn  —  an~^X. 

y1 

Ans.  nx,  — . 
nx 

2.  Find  the  subtangent  to  tbe  curve 

%myii  —  am+n. 

nx 

Ans. . 

m 

3.  Find  the  equation  of  the  tangent  to  the  curve 

x5  =  a?y*. 

Ans. 

=3- 

x         y 

4.  Show  that  the  points  of  contact  of  tangents  from  a  point  (a,  /3)  to  the 
curve 

xmyn  =  am+n 

are  situated  on  the  hyperbola  (m  +  n)xy  =  nfix  +  may. 

5.  In  the  same  curve  prove  that  the  portion  of  the  tangent  intercepted  be- 
tween the  axes  is  divided  at  its  point  of  contact  into  segments  which  are  to  each 
other  in  a  constant  ratio. 

6.  Find  the  equation  of  the  tangent  at  any  point  to  the  hypocycloid,  #!  +  y% 
—  a$ ;  and  prove  that  the  portion  of  the  tangent  intercepted  between  the  axes  is 
of  constant  length. 

7.  In  the  curve  X"-  +  yn  =  an,  find  the  length  of  the  perpendicular  drawn 
from  the  origin  to  the  tangent  at  any  point,  and  find  also  the  intercept  made  by 
the  axes  on  the  tangent. 

an  .  ,  .  n2n 

Ans.  p  =  —  ;  intercept  = 


v/^n-Z  +  yZn-l'  px^yn-^ 

8.  If  the  co-ordinates  of  every  point  on  a  curve  satisfy  the  equations 

x  =  c  sin  20 (1  +  cos  26),     y  =  c  cos  20  (1  —  cos  20), 
prove  that  the  tangent  at  any  point  makes  the  angle  6  with  the  axis  of  x. 

9.  The  co-ordinates  of  any  point  in  the  cycloid  satisfy  the  equations 

x  =  a(9  —  sin0),     y  =  a(i  —  cos  6): 
prove  that  the  angle  which  the  tangent  at  the  point  makes  with  the  axis  of  y 


is  -. 

2 


Examples.  239 

Here  -f  =  -7-  =  cot  -. 

ax      ax  2 

dd 

10.  Prove  that  the  locus  of  the  foot  of  the  perpendicular  from  the  pole  on 
the  tangent  to  an  equiangular  spiral  is  the  same  curve  turned  through  an  angle. 

11.  Prove  that  the  reciprocal  polar,  with  respect  to  the  origin,  of  an  equi- 
angular spiral  is  another  spiral  equal  to  the  original  one. 

12.  An  equiangular  spiral  touches  two  given  lines  at  two  given  points ;  prove 
that  the  locus  of  its  pole  is  a  circle. 

13.  Find  the  equation  of  the  reciprocal  polar  of  the  curve 

rk  cos  -  =  aK 
3 

a 

with  respect  to  the  origin.  Ans.  The  Cardioid  H  =  ah  cos  -. 

14.  Find  the  equation  of  the  inverse  of  a  conic,  the  focus  being  the  pole  of 
inversion. 

15.  Apply  Art.  184,  to  prove  that  the  equation  of  the  inverse  of  an  ellipse 
with  respect  to  any  origin  0  is  of  the  form 

2aP  =  OFi  .  pi  +  OF2  .  p2, 

where  F\  and  F%  are  the  foci,  and  p,  pi,  02  represent  the  distances  of  any  point 
on  the  curve  from  the  points  0,  /1  and  /2,  respectively  ;  f\  and  f%  being  the 
points  inverse  to  the  foci,  F\  and  F%. 

16.  The  equation  of  a  Cartesian  oval  is  of  the  form 

r  +  Jcr'  =  af 

where  r  and  r'  are  the  distances  of  any  point  on  the  curve  from  two  fixed  points, 
and  a,  k  are  constants.  Prove  that  the  equation  of  its  inverse,  with  respect  to 
any  origin,  is  of  the  form 

api  +  j3/>2  +  7/>3  =  o, 

where  pi,  p2,  pz  are  the  distances  of  any  point  on  the  curve  from  three  fixed 
points,  and  a,  0,  7  are  constants. 

17.  In  general  prove  that  the  inverse  of  the  curve 

api  +  #/}2  +  yps  =  o, 

with  respect  to  any  origin,  is  another  curve  whose  equation  is  of  similar  form. 

18.  If  the  radius  vector,  OF,  drawn  from  the  origin  to  any  point  Pon  a 


240  Examples. 

curve  be  produced  to  Pi,  until  PPi  be  a  constant  length ;  prove  that  the  normal 
at  Pi  to  the  locus  of  Pi,  the  normal  at  P  to  the  original  curve,  and  the  perpen- 
dicular at  the  origin  to  the  line  OP,  all  pass  through  the  same  point. 

This  follows  immediately  from  the  value  of  the  polar  subnormal  given  in 
Art.  182. 

19.  If  a  constant  length  measured  from  the  curve  be  taken  on  the  normals 
along  a  given  curve,  prove  that  these  lines  are  also  normals  to  the  new  curve 
which  is  the  locus  of  their  extremities. 

20.  In  the  ellipse  —  +  ^-  =  I,  if  #  =  #  sin  d>, 

a-      o% 

prove  that 

ds 


=  a\/ 1  -  e2  sin2d>. 
d$ 

21.  If  ds  be  the  element  of  the  arc  of  the  inverse  of  an  ellipse  with  respect 
to  its  centre,  prove  that 

,    a  \/i  —  e2  sin2  d>  '  ,  a2  -  b2 

ds  =  &2  — ^~-  dtp,    where  n  =  — - — . 

b2        if  « sin2<f>  b* 

22.  If  co  be  the  angle  which  the  normal  at  any  point  on  the  ellipse 

— \.  —  =  1  makes  with  the  axis-major,  prove  that 
a2      b2 

b2  da 

ds  = 


a  (1  —  £2sin3w)2 


23.  Express  the  differential  of  an  elliptic  arc  in  terms  of  the  semi -axis  major, 
fi,  of  the  confocal  hyperbola  which  passes  through  the  point. 


I  a2  -  a2 


24.  In  the  curve  rm  =  am  cos  mB,  prove  that 

T  »n-l 

ds  „ 

-—  =  a  sec  m  m9. 
dO 

25.  If  F(x,  y)  —  o  be  the  equation  to  any  plane  curve,  and  <p  the  angle  be- 
tween the  perpendicular  from  the  origin  on  the  tangent  and  the  radius  vector  to 
the  point  of  contact,  prove  that 

dF        dF 
dx         dy 

tm*  =  — dF' 

dx        dy 


(     24i      ) 


CHAPTEE  XIII. 

ASYMPTOTES. 

196.  Intersection  of  a   Curve   and  a  Right  Line. — 

Before  entering  on  the  subject  of  this  chapter  it  will  be  ne- 
cessary to  consider  briefly  the  general  question  of  the  inter- 
section of  a  right  line  with  a  curve  of  the  ntn  degree. 

Let  the  equation  of  the  right  line  be  y  =  \ix  +  v,  *and  sub- 
stitute fix  +  v  instead  of  y  in  the  equation  of  the  curve ;  then 
the  roots  of  the  resulting  equation  in  x  represent  the  abscissae 
of  the  points  of  section  of  the  line  and  curve. 

Moreover,  as  this  equation  is  always  of  the  nth  degree,  it 
follows  that  every  right  line  meets  a  curve  of  the  nth  degree  in  n 
points,  real  or  imaginary,  and  cannot  meet  it  in  more. 

If  two  roots  in  the  resulting  equation  be  equal,  two  of  the 
points  of  section  become  coincident,  and  the  line  becomes  a 
tangent  to  the  curve. 

Again,  suppose  the  equation  of  the  curve  written  in  the 
form  of  Art.  175,  viz. : 

Un  +  Un-l  +  W«_2  +  .  .   .  U2  +  U\  +  UQ  =  O  J 

then,  since  un  is  a  homogeneous  function  of  the  nth  degree  in 
x  and  y,  it  can  be  written  in  the  form  xnfA-\\  similarly 

u^^x^ffi),    un_2  =  xn~*fz(y),&Q. 
And  accordingly,  the  equation  of  the  curve  may  be  written, 

*»/„Q  +  *-yi(§)  +  a"-y*(J) + &c.  =  o.       (1) 

v        y 
Substituting  ju  +  -  f  or  -  in  this,  it  becomes 
x        x 

xn/°  (^ + % + x"'i/i  {,i+t)+ ***  (m + *) + &°- = °* 

R 


242  Asymptotes. 

Or,  expanding  by  Taylor's  Theorem, 

*/.G»)  +  «M  { v/,W  +/,  (/*) }  +  ^2  { £  v2/o"M  +  v//(M)  +/.W ) 

+  &C.  =  O.  (2) 

The  roots  of  this  equation  determine  the  points  of  section  in 
question. 

We  add  a  few  obvious  conclusions  from  the  results  arrived 
at  above : — 

i°.  Every  right  line  must  intersect  a  curve  of  an  odd  de- 
gree in  at  least  one  real  point ;  for  every  equation  of  an  odd 
degree  has  one  real  root. 

20.  A  tangent  to  a  curve  of  the  nth  degree  cannot  meet  it 
in  more  than  n  -  2  points  besides  its  points  of  contact. 

3°.  Every  tangent  to  a  curve  of  an  odd  degree  must  meet 
it  in  one  other  real  point  besides  its  point  of  contact. 

40.  Every  tangent  to  a  curve  of  the  third  degree  meets 
the  curve  in  one  other  real  point. 

197.  Definition  of  an  Asymptote. — An  asymptote  is 
a  tangent  to  a  curve  in  the  limiting  position  when  its  point 
of  contact  is  situated  at  an  infinite  distance. 

i°.  No  asymptote  to  a  curve  of  the  nth  degree  can  meet  it 
in  more  than  n  -  2  points  distinct  from  that  at  infinity. 

2°.  Each  asymptote  to  a  curve  of  the  third  degree  inter- 
sects the  curve  in  one  point  besides  that  at  infinity. 

198.  Method  of  finding  the  Asymptotes  to  a  Curve 
of  the  nth  Degree. — If  one  of  the  points  of  section  of  the 
line  y  =  fix  +  v  with  the  curve  be  at  an  infinite  distance,  one 
root  of  equation  (2)  must  be  infinite,  and  accordingly  we 
have  in  that*  case 

/oW  =  o.  (3) 

Again,  if  two  of  the  roots  be  infinite,  we  have  in  addition 

vfiip)  +/i(ju)  =  o.  (4) 

*  This  can  be  easily  established  by  aid  of  the  reciprocal  equation ;  for  if  we 

substitute  -  for  x  in  equation  (2),  the  resulting  equation  in  z  will  have  one  root 

z 

zero  wnen  its  absolute  term  vanishes,  i.e.,  when /o (ft)  =  o;  it  has  two  roots 
zero  when  we  have  in  addition  v/o'(m)  +  /i(/*)  =  °  >  an^  so  on. 


Method  of  finding  Asymptotes  in  Cartesian  Co-ordinates.  243 

Accordingly,  when  the  values  of  \x  and  v  are  determined 
so  as  to  satisfy  the  two  preceding  equations,  the  correspond- 
ing line 

y  =  fix  +  v 

meets  the  curve  in  two  points  in  infinity,  and  consequently  is 
an  asymptote.     (Salmon's  Conic  Sections,  Art.  154-) 

Hence,  if  fix  be  a  root  of  the  equation  f0(fi)  =  o,  the  line 

y  =  n&-  777— x  \5) 

Jo  (fli) 

is  in  general  an  asymptote  to  the  curve. 

If  /i(ju)  =  o  and/0(ju)  =  o  have  a  common  root  (jui  suppose), 
the  corresponding  asymptote  in  general  passes  through  the 
origin,  and  is  represented  by  the  equation 

y  =  ti&. 

In  this  case  un  and  un^  evidently  have  a  common  factor. 

The  exceptional  case  when'/0'(ju)  vanishes  at  the  same 
time  will  be  considered  in  a  subsequent  Article. 

To  each  root  of  f0(/i)  =  o  corresponds  an  asymptote,  and 
accordingly,*  every  curve  of  the  nth  degree  has  in  general  n 
asymptotes,  real  or  imaginary. 

From  the  preceding  it  follows  that  every  line  parallel 
to  an  asymptote  meets  the  curve  in  one  point  at  infinity. 
This  also  is  immediately  apparent  from  the  geometrical 
property  that  a  system  of  parallel  lines  may  be  considered 
as  meeting  in  the  same  point  at  infinity — a  principle  intro- 
duced, by  Desargues  in  the  beginning  of  the  seventeenth 
century,  and  which  must  be  regarded  as  one  of  the  first 
important  steps  in  the  progress  of  modern  geometry. 

Con.  No  line  parallel  to  an  asymptote  can  meet  a  curve 
of  the  nth  degree  in  more  than  (n  —  i)  points  besides  that 
at  infinity. 

Since  every  equation  of  an  odd  degree  has  one  real 
root,  it  follows  that  a  curve  of  an  odd  degree  has  one  real 


*  Since /o(a0  is  of  the  nth  degree  in  jx,  unless  its  highest  coefficient  vanishes, 
in  which  case,  as  we  shall  see,  there  is  an  additional  asymptote  parallel  to  the  axis 
of'jf.  '      • 

R   2 


244  Asymptotes. 

asymptote,  at  least,  and  has  accordingly  an  infinite  branch 
or  branches.  Hence,  no  curve  of  an  odd  degree  can  be  a  closed 
curve. 

For  instance,  no  curve  of  the  third  degree  can  be  a  finite 
or  closed  curve. 

The  equation  fo(fi)  =  o,  when  multiplied  by  xn,  becomes 
un  =  o ;  consequently  the  n  right  lines,  real  or  imaginary, 
represented  by  this  equation,  are,  in  general,  parallel  to  the 
asymptotes  of  the  curve  under  consideration. 

In  the  preceding  investigation  we  have  not  considered 
the  case  in  which  a  root  of  /0(ju)  =  o  either  vanishes  or  is 
infinite;  i.e.,  where  the  asymptotes  are  parallel  to  either 
co-ordinate  axis.  This  case  will  be  treated  of  separately  in  a 
subsequent  Article. 

If  all  the  roots  of  f0(/.i)  =  o  be  imaginary  the  curve 
has  no  real  asymptote,  and  consists  of  one  or  more  closed 
branches. 

Examples. 

To  find  the  asymptotes  to  the  following  curves  : — 

1.  y%  =  ax2  +  %?. 

Substituting  fix  +  v  for  y,  and  equating  to  zero  the  coefficients  of  x3  and  x\ 
separately,  in  the  resulting  equation,  we  obtain 

Hz  —  i  =  o,  and  iixrv  =  a ; 

a 

hence  the  curve  has  but  one  real  asymptote,  viz., 

a 

2.  #4  —  a4  +  2ax2y  =  b2x2. 
Here  the  equations  for  determining  the  asymptotes  are 

^-1=0,        and  4[xzv  +  2afi  =  o ; 
accordingly,  the  two  real  asymptotes  are 

a        ,  a 

y  =  x  -  r,  and  y  +  x  +  -  =  o. 

3.  xz  +  $x2y  -  xy2  -  zy3  +  x2  -  2xy  +  w2  +  $x  +  5  =  o. 

x      3  I  3 

Ans.  y  +  -  +  -  =  o,    y  =  x+-,    y  +  x=-. 

3      4  4  * 


Asymptotes  Parallel  to  Co-ordinate  Axes,  245 

199.  Case  in  which  un  =  o  represents  the  n  Asymp- 
totes.— If  the  equation  of  the  curve  contain  no  terms  of 
the  (n  -  i)th  degree,  that  is,  if  it  be  of  the  form 

un  +  iin_2  +  Un-z  +  &c   .   .  .  +  Mi  +  uQ  =  o, 

the  equations  for  determining  the  asymptotes  become 

/0(ju)  =  o,  and  vfo'(p)  =  o. 

The  latter  equation  gives  v  =  o,  unless  fo(fj)  vanishes  along 
with/0(ju),  i.e.,  unless /0(ju)  has  equal  roots. 

Hence,  in  curves  whose  equations  are  of  the  above  form, 
the  n  right  lines  represented  by  the  equation  un  =  o  are  the 
n  asymptotes,  unless  two  of  these  lines  are  coincident. 

This  exceptional  case  will  be  considered  in  Art.  202. 

The  simplest  example  of  the  preceding  is  that  of  the 
hyperbola 

ax2  +  2hxy  +  by1  =  c, 

in  which  the  terms  of  the  second  degree  represent  the  asymp- 
totes (Salmon's  Conic  Sections,  Art.  195). 

Examples. 

Find  the  real  asymptotes  to  the  curves 

1.  xy2  -  x^y  =  ar(x  +  y)  +  bz.  Ans.  x  =  o,  y  =  o,  x  -y  -o. 

2.  yz  —  x3  =  a?x.  „  y  —  x  =  O. 

3.  #4  -  yi  =  a2xy  +  b2y2.  „  x  +  y  =  o,  x  -  y  =  o. 

200.    Asymptotes     parallel     to     the     Co-ordinate 

Axes. — Suppose  the  equation  of  the  curve  arranged  accord- 
ing to  powers  of  x,  thus 

a0xn  +  {axy  +  h)xn~l  +  &c.  =  o  ; 

then,  if  a0  =  o  and  axy  +  b  =  o,  or  y  = ,  two  of  the  roots 

#1 

of  the  equation  in  x  become  infinite ;  and  consequently  the 

line  a$  +  b  =  o  is  an  asymptote. 


246  Asymptotes. 

In  other  words,  whenever  the  highest  power  of  x  is 
wanting  in  the  equation  of  a  curve,  the  coefficient  of  the 
next  highest  power  equated  to  zero  represents  an  asymptote 
parallel  to  the  axis  of  x. 

If  a0  =  o,  and  b  =  o,  the  axis  of  x  is  itself  an  asymptote. 

If  xn  and  xn~x  be  both  wanting,  the  coefficient  of  xn~2  re- 
presents a  pair  of  asymptotes,  real  or  imaginary,  parallel  to 
the  axis  of  x  ;  and  so  on. 

In  like  manner,  the  asymptotes  parallel  to  the  axis  of  y 
can  be  determined. 

Examples. 

Find  the  real  asymptotes  in  the  following  curves  : — 

1.  y2x  —  ay2  =  x3  +  ax2  +  b3.  Am.  x  =  a,  y  =  x  +  a,  y  +  x  +  a  =  o. 

2.  y{x2  -  $bx  +  2b2)  =  x3  —  3««2  +  a3,     x  =  b,  x  =  2b,  y  +  ia  =  x  +  $b. 

3.  x^y2  =  a2 (x2  +  y2).  x  =  ±a,  y  =  ±a. 

4.  x2y2  =  a2(x2  —  ?/2).  y  4-  a  =  o,  y  —  a  =  o. 

5.  y2a  —  y2x  =  x3.  x  =  a. 

201.  Parabolic  Branches. — Suppose  the  equation 
/o(/0  =  o  has  equal  roots,  then/0'(jUi)  vanishes  along  with/o(At), 
and  the  corresponding  value  of  v  found  from  (5)  becomes  in- 
finite, unless  /i(/u)  vanish  at  the  same  time. 

Accordingly,  the  corresponding  asymptote  is,  in  general, 
situated  altogether  at  infinity. 

The  ordinary  parabola,  whose  equation  is  of  the  form 

(ax  +  j3?/)2  =  lx  +  my  +  n, 

furnishes  the  simplest  example  of  this  case,  having  the 
line  at  infinity  for  an  asymptote.  (Salmon's  Conic  Sections, 
Art.  254.) 

Branches  of  this  latter  class  belonging  to  a  curve  are 
called  parabolic,  while  branches  having  a  finite  asymptote  are 
called  hyperbolic. 

202.  From  the  preceding  investigation  it  appears  that 
the  asymptotes  to  a  curve  of  the  nth  degree  depend,  in 
general,  only  on  the  terms  of  the  11th  and  the  (n  -  i)th  degrees 


Parallel  Asymptotes.  247 

in  its  equation.  Consequently,  all  carves  which  have  the 
same  terms  of  the  two  highest  degrees  have  generally  the  same 
asymptotes. 

There  are,  however,  exceptions  to  this  rule,  one  of  which 
will  be  considered  in  the  next  Article. 

203.  Parallel  Asymptotes. — We  shall  now  consider 
the  case  where  fo(ji)  ~  o  has  a  pair  of  equal  roots,  each  repre- 
sented by  jui,  and  where  /i(jui)  =  o,  at  the  same  time. 

In  this  case  the  coefficients  of  xn  and  xn~x  in  (2)  both 
vanish  independently  of  v,  when  jit  =  jUi  ;  we  accordingly 
infer  that  all  lines  parallel  to  the  line  y  =  fxix  meet  the  curve 
in  two  points  at  infinity,  and  consequently  are,  in  a  certain 
sense,  asymptotes.  There  are,  however,  two  lines  which  are 
more  properly  called  by  that  name  ;  for,  substituting  jui  for  fi 
in  (2),  the  two  first  terms  vanish,  as  already  stated,  and  the 
coefficient  of  xn~%  becomes 

Hence,  if  vi  and  v%  be  the  roots  of  the  quadratic 


v3 


2/o"W+v//W+/2W=o),  (6) 

the  lines  y  =  /mix  +  vi}  and  y  =  \xxx  +  v2, 

are  a  pair  of  parallel  asymptotes,  meeting  the  curve  in  three 
points  at  infinity. 

If  the  roots  of  the  quadratic  be  imaginary,  the  corre- 
sponding asymptotes  are  also  imaginary. 

Again,  if  the  term  un^  be  wanting  in  the  equation,  and 
if  /0(jtt)  =  o  have  equal  roots,  the  corresponding  asymptotes 
are  given  by  the  quadratic 


— /o"(/*l)  +/8(jUi)  =  o. 

In  order  that  these  asymptotes  should  be  real,  it  is 
necessary  that/2(jKi)  and  /o"(/*i)  should  have  opposite  signs. 

There  is  no  difficulty  in  extending  the  preceding  investi- 
gation to  the  case  where  fo(ji)  =  o  has  three  or  more  equal 
roots. 


248  Asymptotes. 


Examples. 

1.  (x  +  yf  {x2  +  y2  +  xy)  =  a2y2  +  a3(x  -  y). 

Here  Mix)  =  (1  +  ^)2(i  +  fx  +  /*«),    /^>  =  o,    /2(M)=-«V; 

.'.  yui  =  —  I,     fo"(m)  =  2,    /2(mi)  =  -«2; 
accordingly  yi  =  «,     y2  =  -  a, 

and  the  corresponding  asymptotes  are 

y  +  x  —  a  =  o,  and  3/  +  x  +  a  =  o. 
The  other  asymptotes  are  evidently  imaginary. 

2.  x2  (x  +  y)2  +  2ay2(x  +  y)  +  Sofixy  +  a3y  =  o. 
Here           fM  =  (1  +  tf,    ffa)  -  2^(1  +  fi),    M/i)  =  Sa^; 

.-.    ^1  =  -  I,      /o"0)  =  2>      /iVl)  =  2«,      /2(/*l)  =  -   8«*> 

and  the  corresponding  asymptotes  are 

y  +  a;  —  2.a  =  o,  and  y  +  #  +  40  =  o. 

204.  If  the  equation  to  a  curve  of  the  nth  degree  be  of 
the  form 

t  (y  +  ax  +  |3)  fa  +  02  =  o, 

where  the  highest  terms  containing  x  and  y  in  fa  are  of  the 
degree  n  -  1 ,  and  those  in  $2  are  of  the  degree  w  -  2  at  most, 
the  line 

y  +  ax  +  |3  =  o 

is  an  asymptote  to  the  curve. 

For,  on  substituting  -  ax  -  j3  instead  of  y  in  the  equation, 
it  is  evident  that  the  coefficients  of  xn  and  xn~l  both  vanish ; 
hence,  by  Art.  198,  the  line  y  +  ax  +  f3  =  o  is  an  asymptote. 

Conversely,  it  can  be  readily  seen  that  if  y  +  ax  +  fi  be  an 
asymptote  to  a  curve  of  the  nth  degree  its  equation  admits  of 
being  thrown  into  the  preceding  form. 

In  general,  if  the  equation  to  a  curve  of  the  nth  degree 
be  of  the  form 

(y  +  axx  +  (3i)  {y  +  a2x  +  j32)  .  .  .  (y  +  anx  +  fdn)  +  fa  =  o,     (7) 


^ 


Asymptotes  to  a  Cubic.  249 

where  <p2  contains  no  term  higher  than  the  degree  n  -  2,  the 
lines 

y  +  axx  +  /3i  =  o,     y  +  a2x  +  j32  =  o,  .   .  .  y  +  anx  +  (5n  =  o 

are  the  n  asymptotes  of  the  curve. 

This  follows  at  once  as  in  the  case  considered  at  the  com- 
mencement of  this  Article. 

For  example,  the  asymptotes  to  the  curve 

xy  (x  +  y  +  ai)  (x  +  y  +  a%)  +  bxx  +  b2y  =  o 
are  evidently  the  four  lines 

x  =0,  y  =  o,  x  +  y  +  ay  =  o,  x  +  y  +  a2  =  o. 

If  the  curve  be  of  the  third  degree,  <£2  is  of  the  first,  and 
accordingly  the  equation  of  such  a  curve,  having  three  real 
asymptotes,  may  be  written  in  the  form 

(y  +  aiX  +  j3i)  (y  +  a2x  +  j32)  (y  +  azx  +  /33)  +  Ix  +  my  +  n  =  o.   (8) 

Hence  we  infer  that  the  three  points  in  which  the  asymp- 
totes to  a  cubic  meet  the  curve  lie  in  the  same  right  line,  viz., 

Ix  +  my  +  n  =  o. 

The  student  will  find  a  short  discussion  of  a  cubic  with 
three  real  asymptotes  in  Chapter  xviii. 

205.  To  prove  that,  in  general,  the  distance  of  a  point 
in  any  branch  of  a  curve  from  the  corresponding  asymptote 
diminishes  indefinitely  as  its  distance  from  the  origin  increases 
indefinitely. 

If  y  +  ax  +  j3  =  o  be  the  equation  of  an  asymptote,  then, 
as  in  the  preceding  Article,  the  equation  of  the  curve  may  be 
written  in  the  form 

(y  +  ax  +  fi>)  (j)x  =  <£2, 
where  02  is  at  least  one  degree  lower  than  (pi  in  x  and  y. 


A 


250  Asymptotes. 

Hence  y  +  ax  +  /3  =  - , 

and  the  perpendicular  distance  of  any  point  (x0,  y0)  on  the 
curve  from  the  line  y  +  ax  +  f5  =  o  is 

y0  +  ax0  +  (5  1        (fa 

— ,  or 


vA  +  a2  ^/i  +  a~2\0iA 

where  the  suffix  denotes  that  #0  and  y0  are  substituted  for  # 
and  y  in  the  functions  fa  and  02. 

Now,  when  x0  and  2/0  are  taken  infinitely  great,  the  value 
of  the  preceding  fraction  depends,  in  general,  on  the  terms 
of  the  highest  degree  (in  x  and  y)  in  fa  and  fa ;  and  since  the 
degree  of  fa  is  one  loiver  than  that  of  fa,  it  can  be  easily 

seen  by  the  method  of  Ex.  7,  Art.  89,  that  the  fraction  — 

0i 
becomes,  in  general,  infinitely  small  when  x  and  y  become 
infinitely  great.  Hence,  the  distance  of  the  line  y  +  ax  +  j3 
from  the  curve  becomes  infinitely  small  at  the  same  time. 

It  is  not  considered  necessary  to  go  more  fully  into  this 
discussion  here. 

The  subject  of  parabolic  and  other  curvilinear  asymptotes 
is  omitted  as  being  unsuited  to  an  elementary  treatise. 
Moreover,  their  discussion,  unless  in  some  elementary  cases, 
is  both  indefinite  and  unsatisfactory,  since  it  can  be  easily 
seen  that  if  a  curve  has  parabolic  branches,  the  number  of  its 
parabolic  asymptotes  is  generally  infinite.  The  reader  who 
desires  full  information  on  this  point,  as  well  as  the  discussion 
of  the  particular  parabolas  called  osculating,  is  referred  to  a 
paper  by  M.  Pliieker,  in  Liouville's  Journal,  vol.  i.,  p.  229. 

206.  Asymptotes  in  Polar  Co-ordinates. — If  a 
curve  be  referred  to  polar  co-ordinates,  the  directions  of  its. 
points  at  an  infinite  distance  from  the  origin  can  be  in  gene- 
ral determined  by  making  r  =  00,  or  u  =  o,  in  its  equation, 
and  solving  the  resulting  equation  in  9.  The  position  of  the 
asymptote  corresponding  to  any  such  value  of  0  is  obtained 
by  finding  the  length  of  the  corresponding  polar  subtangent, 

i.e.,  by  finding  the  value  of  —  corresponding  to  u  =  o. 

WUi 


Asymptotes  in  Polar  Qo-ordinates.  251 

d9 
It  should  be  observed  that  when  —  is  positive,  the  asymp- 

Cvtv 

tote  lies  above  the  corresponding  radius  vector,  and  when 
negative,  below  it ;  as  is  easily  seen  from  Art.  182. 

If  we  suppose  the  equation  of  the  curve,  when  arranged 
in  powers  of  r,  to  be 

*Vo(0)  +  OW)  +  •  •  •  +  rf**(P)  +  /»(&)  =  o, 

the  transformed  equation  in  u  is 

u%(0)  +  t*"-%-!(0)  +  .  .  .  +  uMO)  +/o(0)  =  o :         (9) 

consequently,  the  directions  of  the  asymptotes  are  given  by 
the  equation 

/o(6)  =  O.  (10) 

Again,  if  we  differentiate  (9)  with  respect  to  0,  it  is  easily 
seen  that  the  values  of  -^  corresponding  to  u  =  o  are  given 
by  the  equation 

A(d)fd  +/.'(«)  =  o,  (11) 

provided  that  none  of  the  functions 

MO),  MB),  .  .  .  MO) 

become  infinite  for  the  values  of  0  which  satisfy  equation  (10). 

Consequently,  if  a  be  a  root  of  the  equation  /o(0)  =  o,  the 

curve  has  an  asymptote  making  the  angle  a  with  the  prime 

vector,  and  whose  perpendicular  distance  from  the  origin  is 

represented  by  -  -jt^t. 

/o(«) 

It  is  readily  seen  that  the  equation  of  the  corresponding 
asymptote  is 

r  sm(a  -  V)  +777—.  =  o. 
Jo  [a) 

This  method  will  be  best  explained  by  applying  it  to  one 
or  two  elementary  Examples. 


252  Asymptotes. 

Examples. 
1.  Let  the  curve  be  represented  by  tbe  equation 


Here  u  = 


r  =  a  sec  0  +  b  tan  0. 

COS0 


a  +  b  sin  0 


when  0  =  -,  we  nave  u  =  o,  and 


2  de      a  +  b 

Accordingly,  the  corresponding  polar  subtangent  is  a  +  b,  and  hence  the  line 
perpendicular  to  the  prime  vector  at  the  distance  a  +  b  from  the  origin  is  an 
asymptote  to  the  curve. 

3"" 
Again,  u  vanishes  also  when  0  =  — ,  and  the  corresponding  value  of  the 

2 

polar  subtangent  is  a  -  b ;  thus  giving  another  asymptote. 
2.  r  =  a  sec  mQ  +  b  tan  md. 

cos  md 


Here 


a  +  b  sin  md 


Txri  ^  1  .  du       —  m 

When  0  =  — ■,  we  have  u  =  o,  and  —  =  = , 

2m  <?0      a  +  b 

whence  we  get  one  asymptote. 

i     •        i                             371"                   ^du        m 
Again,  when  0  =  — ,  u  =  o,  and  —  =  -, 

which  gives  a  second  asymptote. 

5"" 

On  making  0  =  — ,  we  get  a  third  asymptote,  and  so  on. 

It  may  be  remarked,  that  the  first,  third,  .  .  .  asymptotes  all  touch  one 
fixed  circle;  and  the  second,  fourth,  &c,  touch  another. 

3.  Find  the  equations  to  the  two  real  asymptotes  to  the  curve 
r2sin(0  —  a)  +  ar  sin(0  —  2a)  +  a?  =  o. 

Ans.  r  sin  (0  -  a)  =  ±  a  sin  a. 

207.  Asymptotic  Circles. — In  some  curves  referred  to 
polar  co-ordinates,  when  6  is  infinitely  great  the  value  of  r 
tends  to  a  fixed  limiting  value,  and  accordingly  the  curve 


Asymptotic  Circles.  253 

approaches  more  and  more  nearly  to  the  circular  form  at  the 
same  time :  in  such  a  case  the  curve  is  said  to  have  a  circular 
asymptote. 

For  example,  in  the  curve 

ad 


0  +  a' 


so  long  as  0  is  positive  r  is  less  than  a,  a  being  supposed 
positive;  but  as  9  increases  with  each  revolution,  r  con- 
tinually increases,  and  tends,  after  a  large  number  of  revo- 
lutions, to  the  limit  a ;  hence  the  circle  described  with  the 
origin  as  centre,  and  radius  a,  is  asymptotic  to  the  curve, 
which  always  lies  inside  the  circle  for  positive  values  of  0. 
Again,  if  we  assign  negative  values  to  0,  similar  remarks  are 
applicable,  and  it  is  easily  seen  that  the  same  circle  is  asymp- 
totic to  the  corresponding  branch  of  the  curve  ;  with  this 
diif erence,  that  the  asymptotic  circle  lies  within  the  curve  in 
the  latter  case,  but  outside  it  in  the  former.  The  student 
will  find  no  difficulty  in  applying  this  method  to  other 
curves,  such  as 

aO  aO2  a(0  +  cos  6) 

r  =  0  +  sin  0'     T  "  F+T2'     r  =     0  +  sin0    ' 


254  Examples. 

Examples. 

Find  the  equations  of  the  real  asymptotes  to  the  following  curves  : — 
i.  y(a2  -  x2)  =  b2{2x  +  e).        Arts,  y  =  o,  x  +  a  =  o,  x  -  a  =  o. 

2.  #4  -  x2y'2  +  a2x%  +  £4  =  o.  x  +  y  =  o,  x  -  y  =  o,  x  =  o. 

3.  x±  —  x2y2  +  x*  +  y2  -  a2  =  o.         x—i=o,  x+i=o,  x-y  =  o,  x-\-y  =  o. 

4.  (a  +  x)2(b2  -  x2)  =  x2y2.  x  =  o, 

5.  (a  +  x)2(b2  +  x2)  =  x2y2.  x  =  o,  y  =  x  +  a,  y  +  x  +  a  =  o. 

6.  xzy  —  2x2y2  +  xy3  =  a2x2  +  b2y2.  x  =  o,  y  =  o,  x  —  y  =  +  v  a2  +  b2. 

7.  a3  —  4xyz  —  %x2  +  lixy  —  i2y2  -f  8x  +  iy  +  4  =  o. 

^4«s.  .r  +  3  =  o,  x  —  iy  =  o,  x  +  2y  =  6. 

8.  #2g/2  —  «#(#  +  #)2  —  2a2y2  —  a4  =  o.     a;  -f  2a  =  o,  x  -  a  =  o. 

9.  If  the  equation  to  a  curve  of  the  third  degree  be  of  the  form 

uz  +  U\  +  Uo  —  o, 
the  lines  represented  by  u%  =  o  are  its  asymptotes. 

10.  If  the  asymptotes  of  a  cubic  be  denoted  by  0  =  0,  fi  =  o,  7  =  0,  the 
equation  of  the  curve  may  be  written  in  the  form 

afiy  =  AaJr£P+  Cy. 

11.  In  the  logarithmic  curve 

X 

y  =  ab, 

prove  that  the  negative  side  of  the  axis  of  x  is  an  asymptote. 

12.  Find  the  asymptotes  to  the  curve 

r  cos  nQ  =  a. 

1 3 .  Find  the  asymptotes  to 

r  cos  mO  =  a  cos  nQ. 

14.  Show  that  the  curve  represented  by 

x3  +  aby  —  axy  =  o 

has  a  parabolic  asymptote,  x2  +  bx  +  b2  =  ay. 


Examples.  255 

15.  Find  the  circular  asymptote  to  the  curve 

ad  +  b 

r  = . 

0  t   a 

16.  Find  the  condition  that  the  three  asymptotes  of  a  cuhic  should  pass 
through  a  common  point. 

Let  the  equation  of  the  curve  be  written  in  the  form 

(to  +  ?>h%  +  3%  *t-  3c0x2  -f  dc\xy  +  l^y1  +  dQx3  +  id\.x2y  +  id^xy2  +  <ky*  =  o, 

then  the  condition  is 

do,         di,         $2, 

d\,        d%,         dz, 

00,  01,  02, 

This  result  can  he  easily  arrived  at  by  substituting  x  +  a  and  y  +  fi  instead 
of  x  and  y  in  the  equation  of  the  cubic,  and  finding  the  condition  that  the  part 
of  the  second  degree  in  the  resulting  equation  should  vanish.     See  Art.  204. 

17.  "When  the  preceding  condition  is  satisfied  show  that  the  co-ordinates, 
a  and  £,  of  the  point  of  intersection  of  the  three  asymptotes,  are  given  by  the 
equations 

e\d\  —  cod2  C()d\  —  c\do 

dodz  —  d±z '  djjdz  —  d^ 

18.  If  from  any  point,  0,  a  right  line  be  drawn  meeting  a  curve  of  the  nth 
degree  in  Mi,  JR2,  •  •  •  Mn,  and  its  asymptotes  in  n,  r%,  .  .  .  rn,  prove  that 

0.fti  +  OB2+  .  .  .  OEu  =  On  +  On  +  .  .  .  Orn. 

N.B. — The  terms  of  the  nth  and  (n  -  \)th  degrees  are  the  same  for  a  curve 
and  its  asymptotes. 

19.  If  a  right  line  be  drawn  through  the  point  (a,  0)  parallel  to  the  asymptote 
of  the  cubic  (x  —  a)3  —  x2y  =  o,  prove  that  the  portion  of  the  line  intercepted  by 
the  axes  is  bisected  by  the  curve. 

20.  If  from  the  origin  a  right  line  be  drawn  parallel  to  any  of  the  asymptotes 
of  the  cubic 

y{ax2  +  zhxy  +  by*  +  igx  +  2fy  +  c)  —  xz  =  o, 

show  that  the  portion  of  this  line  intercepted  between  the  origin  and  the  line 
gx  +fy  +  e  =  o  is  bisected  by  the  curve. 

21.  If  tangents  be  drawn  to  the  curve  xz  +  yz  =  a3  from  any  point  on  the 
line  y  =  x,  prove  that  their  points  of  contact  lie  on  a  circle. 

22.  Show  that  the  asymptotes  to  the  cubic 

ax2y  +  bxy2  +  a'x2  +  b'y2  +  a'x  +  b"y  =  o 
are  always  real,  and  find  their  equations. 

Am.         bx  +  V  =  o,     ay  +  a'  =  o, 
ab  (ax  +  by)  —  a2b'  -  a' b2  =  o. 


(     256    ) 


CHAPTER  XIV. 

MULTIPLE    POINTS    ON    CURVES. 

208.  In  the  following  elementary  discussion  of  multiple 
points  of  curves  the  method  given  by  Dr.  Salmon  in  his 
Higher  Plane  Curves  has  been  followed,  as  being  the  simplest, 
and  at  the  same  time  the  most  comprehensive  method  for 
their  investigation.  The  discussion  here  is  to  be  regarded  as 
merely  introductory  to  the  more  general  investigation  in  that 
treatise,  to  which  the  student  is  referred  for  fuller  information 
on  this  as  well  as  on  the  entire  theory  of  curves. 

"We  commence  with  the  general  equation  of  a  curve  of  the 
nth  degree,  which  we  shall  write  in  the  form 

a0 

+  b0x  +  bxy 

+  c$>2  +  Cixy  +  c2y* 

+    &c.  +  &c. 

+  l(&n  +  lxxn-hj  +  &g.    +  lnyn  =  o, 

where  the  terms  are  arranged  according  to  their  degrees  in 
ascending  order. 

When  written  in  the  abbreviated  form  of  Art.  175,  the 
preceding  equation  becomes 

U0  +  Ux  +  U%  +  .  .  .  +  %_i  +  Un  =  O. 

"We  commence  with  the  equation  in  its  expanded  shape, 
and  suppose  the  axes  rectangular.     Transforming  to  polar 


Multiple  Points.  257 

co-ordinates,  by  substituting  r  cos  0  and  r  sin  0  instead  of 
x  and  y,  we  get 

a0  +  (#0  cos  0  +  #1  sin  0)  r 

+  (tf0  cos20  +  Ci  cos  0  sin  0  +  c2  sin20)  r2  +  .  .  . 

+  (l0cosn0  +  h cosT1 6  sin0  +  .  .  .  +  4sin"0)rn  =  o.      (1) 

If  0  be  considered  a  constant,  the  n  roots  of  this  equation 
in  r  represent  the  distances  from  the  origin  of  the  n  points 
of  intersection  of  the  radius  vector  with  the  curve. 

If  a0  =  o,  one  of  these  roots  is  zero  for  all  values  of  0;  as 
is  also  evident  since  the  origin  lies  on  the  curve  in  this  case. 

A  second  root  will  vanish,  if,  besides  a0  =  o,  we  have 
b0  cos  0  +  bx  sin  0  =  o.  The  radius  vector  in  this  case  meets 
the  curve  in  two  consecutive  points*  at  the  origin,  and  is 
consequently  the  tangent  at  that  point. 

The  direction  of  this  tangent  is  determined  by  the 
equation 

bQ  cos  0  +  h  sin  0  =  o ; 

accordingly,  the  equation  of  the  tangent  at  the  origin  is 

b0x  +  biy  =  o. 

Hence  we  conclude  that  if  the  absolute  term  be  wanting 
in  the  equation  of  a  curve,  it  passes  through  the  origin,  and 
the  linear  part  (u^  in  its  equation  represents  the  tangent  at 
that  point. 

If  b0  =  o,  the  axis  of  x  is  a  tangent ;  if  bx  =  o,  the  axis 
of  y  is  a  tangent. 

The  preceding,  as  also  the  subsequent  discussion,  equally 
applies  to  oblique  as  to  rectangular  axes,  provided  we  sub- 
stitute mr  and  nr  for  x  and  y ;  where 

sin  fa  -  0)         n        sin  0 

m= : -,  sman=- ; 

sm  w  8inw 

to  being  the  angle  between  the  axes  of  co-ordinates. 

From  the  preceding,  we  infer  at  once  that  the  equation  of 
the  tangent  at  the  origin  to  the  curve 

x*  (x2  +  y2)  =  a  (x  -  y) 

*  Two  points  which  are  infinitely  close  to  each  other  on  the  same  branch  of 
a  curve  are  said  to  he  consecutive  points  on  the  curve. 

S 


258  Multiple  Points  on  Curves. 

is  x  -  y  =  o,  a  line  bisecting  the  internal  angle  between  the 
co-ordinate  axes.  In  like  manner,  the  tangent  at  the  origin 
can  in  all  cases  be  immediately  determined. 

209.  Equation  of  Tangent  at  any  Point. — By  aid 

of  the  preceding  method  the  equation  of  the  tangent  at  any 
point  on  a  curve  whose  equation  is  algebraic  and  rational 
can  be  at  once  found.  For,  transferring  the  origin  to  that 
point,  the  linear  part  of  the  resulting  equation  represents  the 
tangent  in  question. 

Thus,  if  f(x,  y)  =  o  be  the  equation  of  the  curve,  we  sub- 
stitute X  +  xx  for  x,  and  Y  +  yx  for  y,  where  (#1,  yi)  is  a 
point  on  the  curve,  and  the  equation  becomes 

f(X  +  wS9  Y  +  yt)  =  o. 

Hence  the  equation  of  the  tangent  referred  to  the  new  axes  is 


\axji         \dyj\ 


On  substituting  x  -  xx,  and  y  -  yx,  instead  of  X  and  Y,  we 
obtain  the  equation  of  the  tangent  referred  to  the  original 
axes,  viz. 

<'-*>(l),+fr-*>(f)>=a 

This  agrees  with  the  result  arrived  at  in  Art.  169. 

210.  Double  Points. — If  in  the  general  equation  of  a 
curve  we  have  a0  =  o,,  bQ  =0,  bx  =  o,  the  coefficient  of  r  is 
zero  for  all  values  of  9,  and  it  follows  that  all  lines  drawn 
through  the  origin  meet  the  curve  in  two  points,  coincident 
with  the  origin. 

The  origin  in  this  case  is  called  a  double  point. 

Moreover,  if  9  be  such  as  to  satisfy  the  equation 

c0  cos20  +  ^cos  9  sin  9  +  c%  sin20  =  o,  (2) 

the  coefficient  of  r2  will  also  disappear,  and  three  roots  of 
equation  ( 1 )  will  vanish. 

As  there  are  two  values  of  tan  9  satisfying  equation  (2),  it 
follows  that  through  a  double  point  two  lines  can  be  drawn, 
each  meeting  the  curve  in  three  coincident  points. 


Double  Points. 


259 


The  equation  (2),  when  multiplied  by  r2,  becomes 

c0x2  +  cxxy  +  c2y2  =  o. 

Hence  we  infer  that  the  lines  represented  by  this  equa- 
tion connect  the  double  point  with  consecutive  points  on  the 
curve,  and  are,  consequently,  tangents  to  the  two  branches  of 
the  curve  passing  through  the  double  point. 

Accordingly,  when  the  lowest  terms  in  the  equation  of  a 
curve  are  of  the  second  degree  (w3),  the  origin  is  a  double 
point,  and  the  equation  u2  =  o  represents  the  pair  of  tangents  at 
that  point. 

For  example,  let  us  consider  the  Lemniscate,  whose  equa- 
tion is 

(x2  +  y2)2  =  a2  (x2  -  y2). 

On  transforming  to  polar  co-ordinates  its  equation  becomes 
ri  =  a2r*  (cos2  0  _  sin20),  or,  r2  =  a2  cos  2O. 

Now,  when  6  =  o,  r  =  ±  a ; 
and,  if  we  confine  our  atten- 
tion to  the  positive  values  of 
r,  we  see  that  as  0  increases 


from   o   to 


7T 


r   diminishes 


from  a  to  zero.     When  0  > 
3*- 


Fig.   18. 


and  <  — ,  r  is  imaginary,  &c, 

and  it  is  evident  that  the  figure  of  the  curve  is  as  annexed, 
having  two  branches  intersecting  at  the  origin,  and  that  the 
tangents  at  that  point  bisect  the  angles  between  the  axes. 
The  equations  of  these  tangents  are 

x  +  y  =  o,  and  x  -  y  =  o, 

results  which  agree  with  the  preceding  theory. 

211.  Wodes,  Cusps,  and  Conjugate  Points.* — The 

pair  of  lines  represented  by  u2  =  o  will  be  real  and  distinct, 
coincident,  or  imaginary,  according  as  the  roots  of  equa- 
tion (2)  are  real  and  unequal,  real  and  equal,  or  imaginary. 

*  These  have  been  respectively  styled  erunodes,  spinodes,  and  acnodes,  by 
Professor  Cayley.     See  Salmon's  Higher  Plane  Curves,  Art.  38. 

S    2 


26o 


Multiple  Points  on  Curves. 


Fig.  19. 


Hence  we  conclude  that  there  may  be  one  of  three  kinds 
of  singular  point  on  a  curve  so  far  as  the  vanishing  of  u0  and  uL 
is  concerned. 

(1).  For  real  and  unequal  roots,  the 
tangents  at  the  double  point  are  real 
and  distinct,  and  the  point  is  called  a 
node;  arising  from  the  intersection  of 
two  real  branches  of  the  curve,  as  in 
the  annexed  figure. 

(2).  If  the  roots  be  equal,  i.e.  if  u2 
be  a  perfect  square,  the  tangents  coin- 
cide, and  the  point  is  called  a  cusp :  the 
two  branches  of  the  curve  touching  each 
other  at  the  point,  as  in  figure  20. 

(3).  If  the  roots  of  u%  be  imaginary, 
the  tangents  are  imaginary,  and  the 
double  point  is  called  a  conjugate  or 
isolated  point ;  the  co-ordinates  of  the  point  satisfy  the  equation 
of  the  curve,  but  the  curve  has  no  real  points  consecutive  to 
this  point,  which  lies  altogether  outside  the  curve  itself. 

It  should  be  observed  also  that  in  some  cases  of  singularities 
of  a  higher  order,  the  origin  is  a  conjugate  point  even  when  u2 
is  a  perfect  square,  as  will  be  more  fully  explained  in  a  sub- 
sequent chapter. 

We  add  a  few  elementary  examples  of  these  different 
classes  for  illustration. 


Fig.  20. 


1. 


Examples. 


Here  the  origin  is  a  node,  the  tangents  bisecting  the  angles  between  the  axes  of 
co-ordinates. 

2.  ay1  =  xz. 

In  this  case  the  origin  is  a  cusp.     Again,  solving  for  y  -we  get 

a* 

Hence,  if  a  be  positive,  y  becomes  imaginary  for  negative  values  of  x ;  and, 
accordingly,  no  portion  of  the  curve  extends  to  the  negative  side  of  the  axis  of  x. 
Moreover,  for  positive  values  of  x,  the  corresponding  values  of  y  have  opposite 
signs.  This  curve  is  called  the  semi-cubical  parabola.  The  form  of  the  curve 
near  the  origin  is  exhibited  in  Fig.  20. 


Double  Points  rn  General.  261 

3.  yz  -  x2  (x  +  a). 

Ans.  The  origin  is  a  cusp. 

4.  b  {x*  +  y2)  =  xK 

Ans.  The  origin  is  a  conjugate  point. 

5.  #3  -  $axy  +  yz  =  o. 

Ans.  The  two  branches  at  the  origin  touch  the  co-ordinate  axes. 

212.  Double  Points  in  Creneral. — In  order  to  seek 
the  double  points  on  any  algebraic  curve,  we  transform  the 
origin  to  a  point  (xly  yi)  on  the  curve  ;  then,  if  we  can  deter- 
mine values  of  xL,  yx  for  which  the  linear  part  disappears  from 
the  resulting  equation,  the  new  origin  (x1}  yx)  is  a  double  point 
on  the  curve. 

From  Art.  209  it  is  evident  that  the  preceding  conditions 
give 

dxjx       '         vW1        ' 

moreover,  since  the  point  (x1}  yx)  is  situated  on  the  curve, 
we  must  have 

/Oi,  tfi)  =  °- 

As  we  have  but  two  variables,  xi9  yif  in  order  that  they 
should  satisfy  these  three  equations  simultaneously,  a  con- 
dition must  evidently  exist  between  the  constants  in  the 
equation  of  the  curve,  viz.,  the  condition  arising  from  the 
elimination  of  a?i,  yt  between  the  three  preceding  equations. 

Again,  when  the  curve  has  a  double  point  (a?i,  «/i),  if  the 
origin  be  transferred  to  it,  the  part  of  the  second  degree  in 
the  resulting  equation  is  evidently 

d2u\      (  dhi  \        2  fdhi\ 


dx'),Jr2xy\dxdyyy  Wk 

Accordingly,  the  lines  represented  by  this  quadratic  are 
the  tangents  at  the  double  point. 

The  point  consequently  is  a  node,  a  cusp,  or  a  conjugate 
point,  according  as 


/  dhi  V  .  fdht\  /WV 

\dxdyjx  \dx2Ji  \dy2  Ji 


262 


Multiple  Points  on  Curves. 


It  may  be  remarked  here  that  no  cubic  can  have  more 
than  one  double  point ;  for  if  it  have  two,  the  line  joining 
them  must  be  regarded  as  cutting  the  curve  in  four  points, 
which  is  impossible. 

Again,  every  line  passing  through  a  double  point  on  a  cubic 
must  meet  the  curve  in  one,  and  but  one,  other  point ;  ex- 
cept the  line  be  a  tangent  to  either  branch  of  the  cubic  at 
the  double  point,  in  which  case  it  cannot  meet  the  curve  else- 
where; the  points  of  section  being  two  consecutive  on  one 
branch,  and  one  on  the  other  branch. 

In  many  cases  the  existence  of  double  points  can  be  seen 
immediately  from  the  equation  of  the  curve.  The  following 
are  some  easy  instances : — 

Examples. 

To  find  the  position  and  nature  of  the   double  points  in  the  following 
curves : — 


1. 


(bx  —  cy)2  =  (x  —  a) 


ab 


The  point  x  =  a,  y  =  — ,  is  evidently  a  cusp, 

G 

at  which  bx  —  cy  =  o  is  the  tangent,  as  in  the 
accompanying  figure 

2.  (y  -  c)2  =  (x  —  #)*  {x  -  b). 

The  point  x  =  a,  y  =  c,  is  a  cusp  if  a  >  b,  or  0 
if  a  =  b  ;  but  is  a  conjugate  point  if  a  <  b. 


Fig.  21. 


3.  yz  =  x(x  +  a)2. 
The  point  y  =  o,  x  =  —  a  is  a.  conjugate  point. 

4.  x§  +  y$  =  «i. 

The  points  x  =  o,  y  =  +  a  ;  and  y  =  o,  x  =  +  a,  are  easily  seen  to  be  cusps. 

213.  Parabolas  of  the  Third  Degree. — The  follow- 
ing example*  will  assist  the  student  towards  seeing  the  dis- 
tinction, as  well  as  the  connexion,  between  the  different  kinds 
of  double  points. 

Let  y2  =  (x  -  a)  (x  -b)  (x  -  c) 

be  the  equation  of  a  curve,  where  a<  b  <  c. 


*  Lacroix,  Gal.  Dif.,  pp.  395-7.     Salmon's  Higher  Plane  Curves,  Art.  39. 


Parabolas  of  the  Third  Degree. 


263 


Here  y  vanishes  when  x  =  a,  ovx=b,  or  x  =  c ;  accordingly, 
if  distances  OA  =  a,  OB  =  b,  OC  =  c,  be  taken  on  the  axis  of 
x,  the  curve  passes  through  the  points  A,  B,  and  C. 

Moreover,  when  x  <  a,  y2  is  negative,  and  therefore 

y  is  imaginary. 
„  x  >  a,  and  <  b,  y2  is  positive,  and  therefore 

y  is  real. 
„  x  >  b,  and  <  c,  y2  is  negative,  and  therefore 

y  is  imaginary. 
„  x  >  c,  y2  is  positive,  and  therefore 

y  is  real ;  and 
increases  indefinitely  along  with  x. 
Hence,  since  the  curve  is  sym- 
metrical with  respect  to  the  axis  of 
x,  it  evidently  consists  of  an  oval 
lying  between  A  and  B,  and  an 
infinite  branch  passing  through 
(7,  as  in  the  annexed  figure.  It 
is  easily  shown  that  the  oval  is 
not  symmetrical  with  respect  to 
the  perpendicular  to  AB  at  its 
middle  point.  Again,  if  b  =  c,  the 
equation  becomes 

y2  =  [x-a){x-  b)2.  Fig.  22. 


In  this  case  the  point  B  co- 
incides with  C,  the  oval  has 
joined  the  infinite  branch,  and 
B  has  become  a  double  point, 
as  in  the  annexed  figure. 


A 


B 


Fig.  23. 


On  the  other  hand,  let  b  =  a,  and  the  equation  becomes 

y2  =  (x  -  a)2  (x  -  c) ; 

in  this  case  the  oval  has  shrunk 
into  the  point  A,  and  the  curve 
is  of  the  annexed  form,  having  0 
A  for  a  conjugate  point. 

Next,  let  a  =  b  =  e,  and  the 
equation  becomes 


A 


y2  =  (x  -  a)s ; 


tfig.  24. 


264  Multiple  Points  on  Curves. 

here  the  points  A,  B,  C,  have 
come  together,  and  the  curve 
has  a  cusp  at  the  point  A,  as  in 
the  annexed  figure. 

The   curves   considered   in 
this  Article  are  called  parabolas  Fig.  25. 

of  the  third  degree. 

As  an  additional  example,  we  shall  investigate  the  fol- 
lowing problem : — 

214.  Given  the  three  asymptotes  of  a  cubic,  to  find  its  equa- 
tion, if  it  have  a  double  point. 

Taking  two  of  its  asymptotes  as  axes  of  co-ordinates,  and 
supposing  the  equation  of  the  third  to  be  ax  +  by  +  c  =  o,  the 
equation  of  the  cubic,  by  Art.  204,  is  of  the  form 

xy[ax  +  by  +  c)  =  Ix  +  my  +  n. 

Again,  the  co-ordinates  of  the  double  point  must  satisfy 
the  equations 

du  du 

dx        '    dy        ' 

or  [lax  +  by  +  c)  y  =  /,       {ax  +  iby  +  c)  x  =  m  ; 

from  which  I  and  m  can  be  determined  when  the  co-ordinates 
of  the  double  point  are  given. 

To  find  n,  we  multiply  the  former  equation  by  x,  and  the 
latter  by  y,  and  subtract  the  sum  from  three  times  the  equa- 
tion of  the  curve,  and  thus  we  get 

cxy  =  zlx  +  2my  +  311 ; 

from  which  n  can  be  found. 

In  the  particular  case  where  the  double  point  is  a  cusp,* 
its  co-ordinates  must  satisfy  the  additional  condition 

dhi  d~u      (  d2u 


dx2  dy*      \dxdy/ 

or  (2  ax  +  iby  +  c)2  =  ^abxy, 

and  consequently  the  cusp  must  lie  on  the  conic  represented 
by  this  equation. 

*  It  is  essential  to  notice  that  the  existence  of  a  cusp  involves  one  more 
relation  among  the  coefficients  of  the  equation  of  a  curve  than  in  the  case  of  an 
ordinary  double  point  or  node. 


Double  Points  on  a  Cubic.  265 

It  can  be  easily  seen  that  this  conic*  touches  at  their 
middle  points  the  sides  of  the  triangle  formed  by  the  asymp- 
totes. 

The  preceding  theorem  is  due  to  Pliicker,f  and  is  stated 
by  him  as  follows  : — 

"  The  locus  of  the  cusps  of  a  system  of  curves  of  the  third 
degree,  which  have  three  given  lines  for  asymptotes,  is  the 
maximum  ellipse  inscribed  in  the  triangle  formed  by  the 
given  asymptotes." 

It  can  be  easily  seen  that  the  double  point  is  a  node  or  a 
conjugate  point,  according  as  it  lies  outside  or  inside  the 
above-mentioned  ellipse. 

215.  Multiple  Points  of  Iliglaer  Curves.— By  follow- 
ing out  the  method  of  Art.  208,  the  conditions  for  the  existence 
of  multiple  points  of  higher  orders  can  be  readily  determined. 

Thus,  if  the  lowest  terms  in  the  equation  of  a  curve  be  of 
the  third  degree,  the  origin  is  a  triple  point,  and  the  tangents 
to  the  three  branches  of  the  curve  at  the  origin  are  given  by 
the  equation  u%  =  o. 

The  different  kinds  of  triple  points  are  distinguished, 
according  as  the  lines  represented  by  u3  =  o  are  real  and 
distinct,  coincident,  or  one  real  and  two  imaginary. 

In  general,  if  the  lowest  terms  in  the  equation  of  a  curve 
be  of  the  mth  degree,  the  origin  is  a  multiple  point  of  the  mth 
order,  &c. 

Again,  a  point  is  a  triple  point  on  a  curve  provided  that 
when  the  origin  is  transferred  to  it  the  terms  below  the  third 
degree  disappear  from  the  equation.  The  co-ordinates  of  a 
triple  point  consequently  must  satisfy  the  equations 

die  du  d2u  d2u  d2u 

9      dx        '     dy       '    dx2        '     dxdy       9     dy2 

Hence  in  general,  for  the  existence  of  a  triple  point  on  a 
curve,  its  coefficients  must  satisfy  four  conditions. 

The  complete  investigation  of  multiple  points  is  effected 

*  From  the  form  of  the  equation  we  see  that  the  lines  x  =  o,  y  =  o  are 
tangents  to  the  conic,  and  that  2ax  +  2by  +  c  =  o  represents  the  line  joining  the 
points  of  contact ;  hut  this  line  is  parallel  to  the  third  asymptote  ax  +  by  +  c  =  o, 
and  evidently  passes  through  the  middle  points  of  the  intercepts  made  hy  this 
asymptote  on  the  two  others. 

t  Zioiwille's  Journal,  vol.  ii.  p.  14. 


266  Multiple  Points  on  Curves. 

more  satisfactorily  by  introducing  the  method  of  trilinear  co- 
ordinates. The  discussion  of  curves  from  this  point  of  view  is 
beyond  the  limits  proposed  in  this  elementary  Treatise. 

215  (a).  Cusps,  in  Greneral. — Thus  far  singular  points 
have  been  considered  with  reference  to  the  cases  in  which 
they  occur  most  simply.  In  proceeding  to  curves  of  higher 
degrees  they  may  admit  of  many  complications  ;  for  instance 
ordinary  cusps,  such  as  represented  in  Fig.  20,  may  be  called 
cusps  of  the  first  species,  the  tangent 
lying  between  both  branches :  the  cases  in 
which  both  branches  lie  on  the  same  side, 
as  exhibited  in  the  accompanying  figure, 
may  be  called  cusps  of  the  second  species.  p.      , 

Professor  Cayley  has  shown  how  this  is 
to  be  considered  as  consisting  of  several  singularities  happen- 
ing at  a  point  (Salmon's  Higher  Plane  Curves,  Art.  58). 

Again,  both  of  these  classes  may  be  called  single  cusps, 
as  distinguished  from  double  cusps  extending  on  both  sides  of 
the  point  of  contact.  Double  cusps  are  styled  tacnodes  by 
Professor  Cayley.  These  points  are  sometimes  called  points 
of  osculation ;  however,  as  the  two  branches  do  not  in  general 
osculate  each  other,  this  nomenclature  is  objectionable.  It 
should  be  observed  that  whenever  we  use  the  word  cusp  with- 
out limitation,  we  refer  to  the  ordinary  cusp  of  the  first  species. 

Cusps  are  calledpoints  de  rebroussement  by  French  writers, 
and  Riickkehrpunkte  by  Grermans,  both  expressing  the  turning 
backwards  of  the  point  which  is  supposed  to  trace  out  the 
curve;  an  idea  which  has  its  English  equivalent  in  their 
name  of  stationary  points.  A  fuller  discussion  of  the  different 
classes  of  cusps  will  be  given  in  a  subsequent  place.  We 
shall  conclude  this  chapter  with  a  few  remarks  on  the  multiple 
points  of  curves  whose  equations  are  given  in  polar  co-ordi- 
nates. 

Examples. 

1.  (y  —  x2)2  =  x5. 
Here  the  origin  is  a  cusp ;  also 

y  =  a;2  +  x%  ; 

hence,  when  a;  is  less  than  unity,  hoth  values  of  y  are  positive,  and  consequently 
the  cusp  is  of  the  second  species. 

2.  Show  that  the  origin  is  a  double  cusp  in  the  curve 

xb  +  bx*  —  a3y2  =  o. 


Multiple  Points  with  Polar  Co-ordinates.  267 

216.  Multiple  Points  of  Curves  iu  Polar  Co-ordi- 
nates.— If  a  curve  referred  to  polar  co-ordinates  pass  through 
the  origin,  it  is  evident  that  the  direction  of  the  tangent  at 
that  point  is  found  by  making  r  =  o  in  its  equation  ;  in  this 
case,  if  the  equation  of  the  curve  reduce  to  f(Q)  =  o,  the 
resulting  value  of  0  gives  the  direction  of  the  tangent  in 
question. 

If  the  equation  f(0)  =  o  has  two  real  roots  in  0,  less  than  71-, 
the  origin  is  a  double  point,  the  tangents  being  determined 
by  these  values  of  0. 

If  these  values  of  9  were  equal,  the  origin  would  be  a  cusp ; 
and  so  on. 

In  fact,  it  will  be  observed  that  the  multiple  points  on 
algebraic  curves  have  been  discussed  by  reducing  them  to 
polar  equations,  so  that  the  theory  already  given  must  apply 
to  curves  referred  to  polar,  as  well  as  to  algebraic  co-ordi- 
nates. 

It  may  be  remarked,  however,  that  the  order  of  a  multiple 
point  cannot,  generally,  be  determined  unless  with  reference 
to  Cartesian  co-ordinates,  in  like  manner  as  the  degree  of  a 
curve  in  general  is  determined  only  by  a  similar  reference. 

For  example,  in  the  equation 

r  =  a  cos20  -  b  sin20, 
the  tangents  at  the  origin  are  determined  by  the  equation 

tan  0  5=  ±    /-,  and  the  origin  would  seem  to  be  only  a  double 

point ;  however,  on  transforming  the  equation  to  rectangular 
axes,  it  becomes 

(a?2  +  y2y  -  (ax2  -  by2)2 ; 

from  which  it  appears  that  the  origin  is  a  multiple  point  of  the 
fourth  order,  and  the  curve  of  the  sixth  degree.  In  fact, 
what  is  meant  by  the  degree  of  a  curve,  or  the  multiplicity  of 
a  point,  is  the  number  of  intersections  of  the  curve  with  any 
right  line,  or  the  number  of  intersections  which  coincide  for 
every  line  through  such  a  point,  and  neither  of  these  are  at 
once  evident  unless  the  equation  be  expressed  by  line  co-ordi- 
nates, such  as  Cartesian,  or  trilinear  co-ordinates;  whereas 
in  polar  co-ordinates  one  of  the  variables  is  a  circular  co- 
ordinate. 


268  Examples. 


Examples. 

i.  Determine  the  tangents  at  the  origin  to  the'curve 

2/2  =  x2  (i  —  x2).         Ans.  x  +  y  =  o,  x  -  y  =  o. 

2.  Show  that  the  curve 

#4  -  $axy  +  2/4  =  o 

touches  the  axes  of  co-ordinates  at  the  origin. 

3.  Find  the  nature  of  the  origin  on  the  curve 

xi  -  ax2y  +  by3  =  o. 

4.  Show  that  the  origin  is  a  conjugate  point  on  the  curve 

ay2  —  xz  +  bx2  =  o 

when  a  and  b  have  the  same  sign  ;  and  a  node,  when  they  have  opposite  signs. 

5.  Show  that  the  origin  is  a  conjugate  point  on  the  curve 

y2  (x2  -  a2)  =  a4. 

6.  Prove  that  the  origin  is  a  cusp  on  the  curve 

7 .  In  the  curve 

(y  —  x2)2  =  xn, 

show  that  the  origin  is  a  cusp  of  the  first  or  second  species,  according  as  n  is 
<  or  >  4. 

8.  Find  the  numher  and  the  nature  of  the  singular  points  on  the  curve 

a:4  +  ^ax3  —  2ay3  +  ^arx2  -  ^a2y2  +  4«4  =  o. 

9.  Show  that  the  points  of  intersection  of  the  curve 


&+  (ff 


with  the  axes  are  cusps. 

10.  Find  the  douhle  points  on  the  curve 

a;4  -  4.CIX3  +  4«2#2  —  b~y2  +  2b3 y  —  a4  -  64  =  O. 


Examples.  269 

11.  Prove  that  the  four  tangents  from  the  origin  to  the  curve 

Mi  +  u%  +  W3  =  o 
are  represented  by  the  equation  4W1  u%  =  u\. 

12.  Show  that  to  a  double  point  on  any  curve  corresponds  an  other  double 
point,  of  the  same  kind,  on  the  inverse  curve  with  respect  to  any  origin. 

13.  Prove  that  the  origin  in  the  curve 

a4  -  2ax2y  —  axy2  +  cfiy2  =  o 

is  a  cusp  of  the  second  species. 

14.  Show  that  the  cardioid 

r  =  «(i  +  cos<?) 

has  a  cusp  at  the  origin. 

15.  If  the  origin  be  situated  on  a  curve,  prove  that  its  first  pedal  passes 
through  the  origin,  and  has  a  cusp  at  that  point. 

16.  Find  the  nature  of  the  origin  in  the  following  curves : — 

ad2 
rz  =  az  sin  %9,  rn  =  an  sin  nd,  r  = . 

0  be  +  c 

17.  Show  that  the  origin  is  a  conjugate  point  on  the  curve 

xi  -  ax2y  +  axy2  +  a2y2  =  o. 

18.  If  the  inverse  of  a  conic  be  taken,  show  that  the  origin  is  a  double  point 
on  the  inverse  curve ;  also  that  the  point  is  a  conjugate  point  for  an  ellipse,  a 
cusp  for  a  parabola,  and  a  node  for  a  hyperbola. 

19.  Show  that  the  condition  that  the  cubic 

xy2  +  ax3  +  bx2  +  ex  +  d  +  ley  =  o 
may  have  a  double  point  is  the  same  as  the  condition  that  the  equation 

ant  +  bx3  +  ex2  +  dx  -  e2  =  0 

may  have  equal  roots. 

20.  In  the  inverse  of  a  curve  of  the  nth  degree,  show  that  the  origin  is  a 
multiple  point  of  the  nth  order,  and  that  the  n  tangents  at  that  point  are  parallel 
to  the  asymptotes  to  the  original  curve. 


270     ) 


CHAPTER  XV. 

ENVELOPES. 

217.  Method  of  Envelopes. — If  we  suppose  a  series  of 
different  values  given  to  a  in  the  equation 

fix,  y,  a)  =  o,  (1) 

then  for  each  value  we  get  a  distinct  curve,  and  the  above 
equation  may  be  regarded  as  representing  an  indefinite 
number  of  curves,  each  of  which  is  determined  when  the 
corresponding  value  of  a  is  known,  and  varies  as  a  varies. 

The  quantity  a  is  called  a  variable  parameter,  and  the 
equation/^,  y,  a)  =  o  is  said  to  represent  a  family  of  curves; 
a  single  determinate  curve  corresponding  to  each  distinct 
value  of  a  ;  provided  a  enters  into  the  equation  in  a  rational 
form  only. 

If  now  we  regard  a  as  varying  continuously,  and  suppose 
the  two  curves 

fix,  y,  a)  =  o,    f[x,  y,a  +  Aa)=0 

taken,  then  the  co-ordinates  of  their  points  of  intersection 
satisfy  each  of  these  equations,  and  therefore  also  satisfy  the 
equation 

f(x,  y,  a  +  Aa)  -fix,  y,  a) 


Aa 


=  o. 


IS  ow,  in  the  limit,  when  Aa  is  infinitely  small,  the  latter 
equa  ion  becomes 

dfix,  y,a)  .  x 

aa 

and  accordingly  the  points  of  intersection  of  two  infinitely 
near  curves  of  the  system  satisfy  each  of  the  equations  (1) 
and  (2). 


Envelopes,  271 

The  locus  of  the  points  of  ultimate  intersection  for  the 
entire  system  of  curves  represented  by  f(x,  y,  a)  =  o,  is  ob- 
tained by  eliminating  a  between  the  equations  (1)  and  (2). 
This  locus  is  called  the  envelope  of  the  system,  and  it  can  be 
easily  seen  that  it  is  touched  by  every  curve  of  the  system. 

For,  if  we  consider  three  consecutive  curves,  and  suppose 
Pi  to  be  the  point  of  intersection  of  the  first  and  second,  and 
P2  that  of  the  second  and  third,  the  line  Pi  P2  joins  two  infi- 
nitely near  points  on  the  envelope  as  well  as  on  the  inter- 
mediate of  the  three  curves  ;  and  hence  is  a  tangent  to  each 
of  these  curves. 

This  result  appears  also  from  analytical  considerations, 
thus  : — the  direction  of  the  tangent  at  the  point  x,  y,  to  the 
curve  /(a?,  y,  a)  =  o,  is  given  by  the  equation 

dx     dy  dx        ' 

in  which  a  is  considered  a  constant. 

Again,  if  the  point  x,  y  be  on  the  envelope,  since  then  a 
is  given  in  terms  of  x  and  y  by  equation  (2),  the  direction  of 
the  tangent  to  the  envelope  is  given  by  the  equation 

df      df  dy      df  (da      da  dy\ 
dx      dy  dx     da  \dx     dy  dx) 

df     df  dy 
dx+dyTx  =  °> 

df 
since  -7-  =  o  for  the  point  on  the  envelope. 
da 

dn 
Consequently,  the  values  of  —  are  the  same  for  the  two 

tlX 

curves  at  their  common  point,  and  hence  they  have  a  common 
tangent  at  that  point. 

One  or  two  elementary  examples  will  help  to  illustrate 
this  theory. 

The  equation  x  cos  a  +  y  sin  a  =  p,  in  which  a  is  a  variable 
parameter,  represents  a  system  of  lines  situated  at  the  same 


272  Envelopes. 

perpendicular  distance  p  from  the  origin,  and  consequently- 
all  touching  a  circle. 

This  result  also  follows  from  the  preceding  theory ;  for 
we  have 

fix,  y,  a)  =  x  cos  a  +  y  sin  a  -  p  =  o, 

df(x,  y,  a) 

— rr- — -  =  -  x  sin  a  +  y  COS  a  =  o, 

da  J  ' 

and,  on  eliminating  a  between  these  equations,  we  get 

x2  +  y2  =  p2y 

which  agrees  with  the  result  stated  above. 
Again,  to  find  the  envelope  of  the  line 

m 
y  =  ax  +  — , 
a 

where  a  is  a  variable  parameter. 

Here  f{x,y,  a)  =  y  -  ax =0, 


a 


df(x,  y,  a)  m 

-  =  —  x  -\     -  =  o ; 

da  a" 


jm 

•*•  a  =  J~~* 

\  x 


Substituting  this  value  for  a,  we  get  for  the  envelope 

y2  =  ^mx, 

which  represents  a  parabola. 

2 1 8.  Envelope  of  La2  +  2Ma  +  JSf=  o.   Suppose  L,  If,  iV, 
to  be  known  functions  of  x  and  y,  and  a  a  parameter,  then 

f(x,  y,  a)  =  La2  +  2Ma  +  N=  O, 

-^  =  zLa  +  2M=  o; 
da 

accordingly,  the  envelope  of  the  curve  represented  by  the 
preceding  expression  is  the  curve 

LN=M2. 


Undetermined  Multipliers  applied  to  Envelopes.         273 

Hence,  when  L,  M,  JSF  are  linear  functions  in  x  and  y, 
this  envelope  is  a  conic  touching  the  lines  L,  iV,  and  having 
M  for  the  chord  of  contact. 

Conversely,  the  equation  to  any  tangent  to  the  conic 
LN  -  M%  can  be  written  in  the  form 

Za2  +  2Ma  +  JSr=0* 

where  a  is  an  arbitrary  parameter. 

219.  Undetermined  Multipliers  applied  to  Enve- 
lopes.— In  many  cases  of  envelopes  the  equation  of  the 
moving  curve  is  given  in  the  form 

f(x,  y,  a,  ($)  =  el9  (3) 

where  the  parameters  a,  ]8  are  connected  by  an  equation  of 
the  form 

0  (a,  P)  =  C2.  (4) 

In  this  case  we  may  regard  j3  in  (3)  as  a  function  of  a  by 
reason  of  equation  (4) ;  hence,  differentiating  both  equations, 
the  points  of  intersection  of  two  consecutive  curves  must 
satisfy  the  two  following  equations : 

df     df  d8  .,  d<f>     d6  dB 

7  +  ^7  =  0,  and  -f  +  -±  -r  =  o. 
da      dp  da  da      dp  da 

d£      d£ 

~  ,,  da      d(5 

Consequently  g-^. 

da      d(5 

If  each  of  these  fractions  be  equated  to  the  undetermined 
quantity  A,  we  get 

da        da 

(5) 
£JT        d$ 

djd        dp 


*  Salmon's  Conies,  Art.  248. 
T 


274  Envelopes. 

and  the  required  envelope  is  obtained  by  eliminating  a,  |3,  and 
X  between  these  and  the  two  given  equations. 

The  advantage  of  this  method  is  especially  found  when 
the  given  equations  are  homogeneous  functions  in  a  and  |3 ; 
for  suppose  them  to  be  of  the  forms 

fix,  y,  a,  (5)  =  d,     <p\a,  0)  =  c2, 

where  the  former  is  homogeneous  of  the  nth  degree,  and  the 
latter  of  the  mth,  in  a  and  ]3.  Multiply  the  former  equation 
in  (5)  by  a,  and  the  latter  by  j3,  and  add ;  then,  by  Euler's 
theorem  of  Art.  102,  we  shall  have 

ncL  =  mc2\,    or  X  =  — -1 ,  (6) 

by  means  of  which  value  we  can  generally  eliminate  a  and  )3 
from  our  equations. 

Examples. 

1.  To  find  the  envelope  of  a  line  of  given  length  (a)  whose  extremities  move 
along  two  fixed  rectangular  axes. 

Taking  the  given  lines  for  axes  of  co-ordinates,  we  have  the  equations 


f  +  y-  =  r,    as  +  £3  =  «2. 

a      £ 

Hence 

x               y 

from  which  we  get 

r 

A.  =  -rj 

a? 

and  the  required  locus  is  represented  by 

#i  +  y%  =  «l. 

2.  To  find  the  envelope  of  a  system  of  concentric  and  coaxal  ellipses  of  con- 
stant area. 

Here  —  4-  —7  =  r ,     afi  —  c; 

x~  y 

hence  - ,  =  Aj8,     -r  =  Ao;    .*.  2\<?=i, 

cr  jS-5 

and  the  required  envelope  is  the  equilateral  hyperbola 

2xy  =  c. 


Examples.  275 

3.  To  find  the  envelope  of  all  the  normals  to  an  ellipse. 
Here  we  have  the  equations 

a2 1  -  p  I  =  «»  -  h\  and  ~  +  fi*  =  1, 
a         ft  a2      b2 

where  a  and  ft  are  the  co-ordinates  of  any  point  on  the  ellipse. 

^  a?z         a      b2y  ft 

Hence,  _  =  A-,    ^  =  -^2; 

consequently  A  =  a2  —  b2, 

and  we  get  a*%  =  (a2  -  b2)  a\    fry  =  -  (a2  -  b2)  ft 3 ; 

a_  /     as    \l      £  _       /     6y     \i 
'''a~  \(fi^7y  '     b~~  [a*  -b2)  ' 

Suhstituting  in  the  equation  of  the  ellipse,  we  get  for  the  required  envelope, 

(axf  +  {by)$={a2-b2)l. 

This  equation  represents  the  evolute  of  the  ellipse. 

x     y 

4.  Find  the  envelope  of  the  line  -  H —  =  1,  where  a  and  ft  are  connected  by 

a      ft 

the  equation 

m  mm 

am  _}_  fim  =  0W,  Ans.  %m+l  +  ym+1  =  cm+l. 

220.  The  preceding  method  can  be  readily  extended  to  the 
general  case  in  which  the  equation  of  the  enveloping  curve 
contains  any  number,  n,  of  variable  parameters,  which  are 
connected  by  n  -  1  independent  equations.  The  method  of 
procedure  is  the  same  as  that  already  considered  in  Chapter 
XI.  on  maxima  and  minima,  and  does  not  require  a  separate 
investigation  here. 


T  2 


276  Examples, 


Examples. 

x      y 
1.  Prove  that  the  envelope  of  the  system  of  lines  —  +  -r-  =  1,  where  I  and  m 

I     m 

7  AV> 

are  connected  by  the  equation  -  +  —  =  1,  is  the  parabola 

■  Of  O 


0H*)'- 


2.  One  angle  of  a  triangle  is  fixed  in  position,  find  the  envelope  of  the 
opposite  side  when  the  area  is  given.  Arts.  A  hyperbola. 

3.  Find  the  envelope  of  a  right  line  when  the  sum  of  the  squares  of  the 
perpendiculars  on  it  from  two  given  points  is  constant. 

4.  Find  the  envelope  of  a  right  line,  when  the  rectangle  under  the  perpen- 
diculars from  two  given  points  is  constant. 

Arts.  A  conic  having  the  two  points  as  foci. 

5.  From  a  point  F  on  the  hypothenuse  of  a  right-angled  triangle,  perpen- 
diculars PM,  FIST  are  drawn  to  the  sides ;  find  the  envelope  of  the  line  MN. 

6.  Find  the  envelope  of  the  system  of  circles  whose  diameters  are  the  chords 
drawn  parallel  to  the  axis-minor  of  a  given  ellipse. 

7.  Find  the  envelope  of  the  circle 

x2  +  y2  -  2aex  +  a2  -  b2  =  o, 

where  a  is  an  arbitrary  parameter ;    and  find  when  the  contact  between  the 
circle  and  the  envelope  is  real,  and  when  imaginary. 

(a).  Show  from  this  example  that  the  focus  of  an  ellipse  may  be  regarded  as 
an  infinitely  small  circle  having  double  contact  with  the  ellipse,  the  directrix 
being  the  chord  joining  the  points  of  contact. 

8.  Show  that  the  envelope  of  the  system  of  conies 

£        V2,    _, 


a      a  —  h 

where  a  is  a  variable  parameter,  is  represented  by  the  equation 

(x  ±  y/h)2  +  y2  =  o. 

Hence  show  that  a  system  of  conies  having  the  same  foci  may  be  regarded 
as  inscribed  in  the  same  imaginary  quadrilateral. 
9.  Find  the  envelope  of  the  line 

xam  +  y&m  =  «m+1, 
where  the  parameters  o  and  /3  are  connected  by  the  equation 
a»»  +  j3n  =  bn. 

n 

Am.  #»-"»  +  yn'm  =  I  -j—  1 


Examples.  277 

10.  On  any  radius  vector  of  a  curve  as  diameter  a  circle  is  described:  prove 
geometrically  that  the  envelope  of  all  such  circles  is  the  first  pedal  of  the  curve 
with  respect  to  the  origin. 

11.  If  circles  be  described  on  the  focal  radii  vectores  of  a  conic  as  diameters, 
prove  that  their  envelope  is  the  circle  described  on  the  axis  major  of  the  conic  as 
diameter. 

12.  Prove  that  the  envelope  of  the  circles  described  on  the  central  radii  of  an 
ellipse  as  diameters  is  a  Lemniscate. 

13.  Find  the  envelope  of  semicircles  described  on  the  radii  of  the  curve 

y»  =  an  cos  nd 
as  diameters. 

14.  If  perpendiculars  be  drawn  at  each  point  on  a  curve  to  the  radii  vectores 
drawn  from  a  given  point,  prove  that  their  envelope  is  the  reciprocal  polar  of 
the  inverse  of  the  given  curve,  with  respect  to  the  given  point. 

15.  Find  the  envelope  of  a  circle  whose  centre  moves  along  the  circum- 
ference of  a  fixed  circle,  and  which  touches  a  given  right  line. 

16.  Ellipses  are  described  with  coincident  centre  and  axes,  and  having  the 
sum  of  their  semiaxes  constant ;  find  their  envelope. 

17.  Find  the  equation  of  the  envelope  of  the  line  \x  +  /xy  +  v  =  o,  where 
the  parameters  are  connected  by  the  equation 

ax*  +  bfi2  +  cv1  +  zffjLV  +  igvX  +  ihXfi  =  o. 


Ans. 


x,        y,        1,        o 

18.  At  any  point  of  a  parabola  a  line  is  drawn  making  with  the  tangent  an 
angle  equal  to  the  angle  between  the  tangent  and  the  ordinate  at  the  point ; 
prove  that  the  envelope  of  the  line  is  the  first  negative  pedal,  with  regard  to  the 

focus,  of  the  parabola ;  and  hence  that  its  equation  is  ri  cos  -  6  =  ai,  the  focus 

3 
being  pole. 

N.B. — This  curve  is  the  caustic  by  reflexion  for  rays  perpendicular  to  the 
axis  of  the  parabola. 

19.  Join  the  centre,  0,  of  an  equilateral  hyperbola  to  any  point,  P,  on  the 
curve,  and  at  P  draw  a  line,  PQ,  making  with  the  tangent  an  angle  equal  to  the 
angle  between  OP  and  the  tangent.  Show  that  the  envelope  of  PQ  is  the  first 
negative  pedal  of  the  curve 

rz  =  2fl2  sin  -  6  sin  -  6, 
3  3 

the  centre  being  pole,  and  axis  minor  prime  vector. 

N.B. — This  gives  the  caustic  by  reflexion  of  the  equilateral  hyperbola,  the 
centre  being  the  radiant  point. 

20.  A  right  line  revolves  with  a  uniform  angular  velocity,  while  one  of  its 
points  moves  uniformly  along  a  fixed  right  line ;  find  its  envelope. 

Ans.  A  cycloid. 


a, 

h, 

9, 

X 

h, 

h 

f, 

y 

ff> 

f, 

c> 

i 

(     278     ) 


OHAPTEE  XVI. 


CONVEXITY  AND  CONCAVITY.      POINTS  OF  INFLEXION. 

221.  Convexity  and  Concavity. — If  the  tangent  be 
drawn  at  any  point  on  a  curve,  the  neighbouring  portion  of 
the  curve  generally  lies  altogether  on  one  side  of  the  tangent, 
and  is  convex  with  respect  to  all  points  lying  at  the  opposite 
side  of  that  line,  and  concave  for  points  at  the  same  side. 

Thus,  in  the  accompanying  figure,  the  portion  QPQ  is 
convex    towards    all    points 
lying  below  the  tangent,  and 
concave  for  points  above. 

If  the  curve  be  referred 
to  the  co-ordinate  axes  OX 
and  OY,  then  whenever  the 
ordinates  of  points  near  to 
P  on  the  curve  are  greater 

than  those  of  the  points  on  N  m  N 

the  tangent  corresponding  to  lg*  27' 

the  same  abscissae,  the  curve  is  said  to  be  concave  towards 
the  positive  direction  of  P. 

Now,  suppose  y  =  <j>  (x)  to  be  the  equation  of  the  curve, 
then  that  of  the  tangent  at  a  point  x,  y,  by  Art.  168,  is 

Let  P  be  the  point  x,  y,  and  MN  =h  =  MN',  QN  =  yly 
TN  =  Pi,  and  we  have 

yx  =  <t>(x  +  h)  =  00)  +  fy'(«)  +  —  *'»  +  »"»  +&c. 


Yx  =  y  +  h(j/(x)  =  (j>(x)  +  h(j)'(x) ; 
...  yx  -  Yl  =  —  <}>"(x)  +  -~^—y\x)  +  &C. 


I  .  2 


1.2.3 


(0 


Points  of  Inflexion. 


279 


When  h  is  very  small,  the  sign  of  the  right-hand  side  of 
this  equation  is  the  same  in  general  as  that  of  its  first  term, 
and  accordingly  the  sign  of  yx  -  TXy  or  of  QT,  is  the  same  as 
that  of  <t>"(x). 

Hence,  for  a  point  ahove  the  axis  of  x,  the  curve  is  convex 
towards  that  axis  when  <l>"(x)  is  positive,  and  concave  when 
negative. 

We  accordingly  see  that  the  convexity  or  concavity  at  any 

point  depends  on  the  sign  of  tf\x)  or  -7^,  at  the  point. 

ttx 

222.  Points  of  Inflexion. — If,  however,  <j>"(%)  =0  at 

the  point  P,  we  shall  have 


ft  -  Tx  = 


h3 


2  .3 


*"»  + 


U 


2.3  .4 


tf*{x)  +  &G.  (2) 


Now,  provided  0"'(a?)  be  not  zero,  ft  -  Tx  changes  its  sign 

with  h,  i.e.  if  MN' =  MN=  h, 

and  if   Q  lies  above  T,  the 

corresponding  point    Q'  lies 

below  T\  and  the  portions  of 

the  curve  near  to   P  lie   at 

opposite  sides  of  the  tangent, 

as  in  the  figure. 

Consequently,  the  tangent 
at  such  a  point  cuts  the  curve, 
as  well  as  touches  it,  at  its  FiS-  28- 

point  of  contact.  Such  points  on  a  curve  are  called  points  of 
inflexion. 

Again,  if  <p'"(x)  as  well  as  $"(%)  vanish  at  the  point  P,  we 
shall  have 

A4 

1  .  2  .  3  . 4  r 

and,  provided  <f>iy(x)  be  not  zero  at  the  point,  yx  -  Tx  does  not 
change  sign  with  A,  and  accordingly  the  tangent  does  not 
intersect  the  curve  at  its  point  of  contact. 

Generally,  the  tangent  does  or  does  not  cut  the  curve  at 
its  point  of  contact,  according  as  the  first  derived  function 
which  does  not  vanish  is  of  an  odd,  or  of  an  even  order  ;  as 
can  be  easily  seen  by  the  preceding  method. 


2  So  Points  of  Inflexion. 

From  the  foregoing  discussion  it  follows  that  at  a  point 
of  inflexion  the  curve  changes  from  convex  to  concave  with 
respect  to  the  axis  of  x,  or  conversely. 

On  this  account  such  points  are  called  points  of  contrary 
flexure  or  of  inflexion. 

223  The  subject  of  inflexion  admits  also  of  being  treated 
by  the  method  of  Art.  196,  as  follows  : — The  points  of  in- 
tersection of  the  line  y  =  fix  +  v  with  the  curve  y  =  $(x)  are 
evidently  determined  by  the  equation 

<p(x)  =  fix  +  v.  (3) 

Suppose  A,  B,  C,  D,  &c,  to  represent  the  points  of  section  in 

question,  and  let  xl9  x2, . . .  xn 

be  the  roots  of  equation  (3) ;  — - yA 

then  the  line  becomes   a   /p 

tangent,   if  two   of  these 

roots   are    equal,    i.e.,    if  Fis-  29- 

<f>'(xi)  =  ju,  where  xx  denotes  the  value  of  x  belonging  to  the 

point  of  contact. 

Again,  three  of  the  roots  become  equal  if  we  have  in 
addition  0"(#i)  =  o  ;  in  this  case  the  tangent  meets  the  curve 
in  three  consecutive  points,  and  evidently  cuts  the  curve  at  its 
point  of  contact ;  for  in  our  figure  the  portions  PA  and  CD 
of  the  curve  lie  at  opposite  sides  of  the  cutting  line,  but 
when  the  points  A,  B,  C  become  coincident,  the  portions  AB 
and  BC  become  evanescent,  and  the  curve  is  evidently  cut  as 
well  as  touched  by  the  line. 

In  like  manner,  if  $m{xi)  also  vanish,  the  tangent  must 
be  regarded  as  cutting  the  curve  in  four  consecutive  points  : 
such  a  point  is  called  a  point  of  undulation. 

It  may  be  observed,  that  if  a  right  line  cut  a  continuous 
branch  of  a  curve  in  three  points,  A,  B,  C,  as  in  our  figure, 
the  curve  must  change  from  convex  to  concave,  or  conversely, 
between  the  extreme  points  A  and  C,  and  consequently  it 
must  have  a  point  of  inflexion  between  these  points  ;  and  so 
on  for  additional  points  of  section. 

Again,  the  tangent  to  a  curve  of  the  nth  degree  at  a  point  of 
inflexion  cannot  intersect  the  curve  in  more  than  n  —  3  other 
points:  for  the  point  of  inflexion  counts  for  three  among 
the  points  of  section.     For  example,  the  tangent  to  a  curve 


Harmonic  Polar  of  a  Point  of  Inflexion  on  a  Cubic.      281 

of  the  third  degree  at  a  point  of  inflexion  cannot  meet  the 
curve  in  any  other  point.  Consequently,  if  a  point  of  in- 
flexion on  a  cubic  be  taken  as  origin,  and  the  tangent  at  it 
as  axis  of  x,  the  equation  of  the  curve  must  be  of  the  form 

#3  +  y$  =  °t 

where  <t>  represents  an  expression  of  the  second  and  lower 
degrees  in  x  and  y.    For,  when  y  =  o,  the  three  roots  of  the 
resulting  equation  in  x  must  be  each  zero,  as  the  axis  of  x 
meets  the  curve  in  three  points  coincident  with  the  origin. 
The  preceding  equation  is  of  the  form 

U3  +  U2  +  Ui  =  o, 

or,  when  written  in  full, 

xz  +  y  (ax2  +  2hxy  +  by2)  +  y  (2gx  +  2fy  +  c)  =  o.       (4) 

Now,  supposing  tangents  drawn  from  the  origin  to  the 
curve,  their  points  of  contact,  by  Art.  176,  lie  on  the  curve 

uz  +  2%h  =  o, 
i.  e.  on  the  curve 

(gx+fy  +  c)y*=o. 

The  factor  y  =  o  corresponds  to  the  tangent  at  the  point 
of  inflexion,  and  the  other  factor  gx  +  fy  +  c  =  o  passes 
through  the  points  of  contact  of  the  three  other  tangents  to 
the  curve. 

Hence,  we  infer  that  from  a  'point  of  inflexion  on  a  cubic 
but  three  tangents  can  be  drawn  to  the  curve,  and  their  three 
points  of  contact  lie  in  a  right  line. 

It  can  be  shown  that  this  right  line  cuts  harmonically 
every  radius  vector  of  the  curve  which  passes  through  the 
point  of  inflexion. 

For,  transforming  equation  (4)  to  polar  co-ordinates,  and 
dividing  by  r9  it  becomes  of  the  form 

Ar2  +  Br  +  C  =  o. 

If  /,  /'  be  the  roots  of  this  quadratic,  we  have 

r  +  r"  ~      C 


282      ft  Points  of  Inflexion. 

Now,  if  p  be  the  harmonic  mean  between  /  and  /',  this 
gives 

2  _  i       i  B         2g  cos  9  +  2/  sin  9 

p     r      r  (J  c 

Hence  the  equation  of  the  locus  of  the  extremities  of  the 
harmonic  means  is 

gx+fy  +  c  =  o.  Q.E.D. 

This  theorem  is  due  to  Maclaurin  {Be  Lin.  Geom.  Prop. 
Gen.,  Sec.  in.  Prop.  9). 

From  this  property  the  line  is  called  the  harmonic  polar  of 
the  point  of  inflexion.  This  line  holds  a  fundamental  place 
in  the  general  theory  of  cubics.* 

224.  Stationary  Tangents. — Since   the  tangent  at  a 

point  of  inflexion  may  be  regarded  as  meeting  the  curve  in 

three  consecutive  points,  it  follows  that  at  such  a  point  the 

tangent  does  not  alter  its  position  as  its  point  of  contact 

passes  to  the  consecutive  point,  and  hence  the  tangent  in  this 

case  is  called  a  stationary  tangent. 

d2y 
The  equation  —  =  o  follows  immediately  from  the  last 

U'X 

consideration ;  for  when  the  tangent  is  stationary  we  must 

have  -j-  =  o,  where  0,   as  in  Art.  171,  denotes  the  angle 
ax 

du 
which  the  tangent  makes  with  the  axis  of  x ;  but  tan  0  =  — , 

ax 

hence  -~  =  o,  which  is  the  same  condition  for  a  point  of 
ax 

inflexion  as  that  before  arrived  at. 


*  Chasles,  Apergu  Historique,  note  xx. ;    Salmon's  Higher  Plane  Curves, 
Art.  179. 


Examples.  283 

Examples. 

1.  Show  that  the  origin  is  a  point  of  inflexion  on  the  cuwe 

a3y  =  bxy  +  cxs  +  dx*. 

2.  The  origin  is  a  point  of  inflexion  on  the  cubic  #3  +  u\  =  o  ? 

3.  In  the  curve  am'xy  =  xm, 

prove  that  the  origin  is  a  point  of  inflexion  if  m  be  greater  than  2. 

4.  In  the  system  of  curves 

yn  =  kxm, 

find  under  what  circumstances  the  origin  is  (a)  a  point  of  inflexion,  (b)  a  cusp. 

5.  Find  the  co-ordinates  of  the  point  of  inflexion  on  the  curve 

2P 
xs  —  3&s2  +  a2y  =  o.  Ans.  x  =»  b,  y  =  —. 

Of 

6.  If  a  curve  of  an  odd  degree  has  a  centre,  prove  that  it  is  a  point  of 
inflexion  on  the  curve. 

7.  Prove  that  the  origin  is  a  point  of  undulation  on  the  curve 

U\  +  Ui  +  «5  +  &c,  +  un  =  o. 

8.  Show  that  the  points  of  inflexion  on  curves  referred  to  polar  co-ordinates 
are  determined  by  aid  of  the  equation 

dHt  ,  1 

«  +  -r-r  =  o,    where  w  =  -. 
ddz  r 

9.  In  the  curve  rdm  =  a,  prove  that  there  is  a  point  of  inflexion  when 

0  —^/m  (1  —  m). 

10.  In  the  curve  y  =  c  sin  -,  prove  that  the  points  in  which  the  curve 
meets  the  axis  of  x  are  all  points  of  inflexion. 

11.  Show  geometrically  that  to  a  node  on  any  curve  corresponds  a  line 
touching  its  reciprocal  polar  in  two  distinct  points  ;  and  to  a  cusp  corresponds  a 
point  of  inflexion. 


284  Examples. 

12.  If  the  origin  be  a  point  of  inflexion  on  the  curve 

U\  +  u%  +  uz  +  .  .  .  +  un  =  o, 

prove  that  u%  must  contain  u\  as  a  factor. 

13.  Show  that  the  points  of  inflexion  of  the  cubical  parabola 

y2  =  (x  -  a)2  (x  -  b) 
lie  on  the  line 

3#  +  a  =  4b : 

and  hence  prove  that  if  the  cubic  has  a  node,  it  has  no  real  point  of  inflexion  ; 
but  if  it  has  a  conjugate  point,  it  has  two  real  points  of  inflexion,  besides  that 
at  infinity. 

14.  Prove  that  the  points  of  inflexion  on  the  curve  y2  =  x2(x2  +  2px  +  q) 
are  determined  by  the  equation  2x3  4  6px2  +  3  (p2  4  q)  x  +  2pq  =  o. 

15.  If  y2  =  f{%)  be  the  equation  of  a  curve,  prove  that  the  abscissae  of  its 
points  of  inflexion  satisfy  the  equation 

{/'W}2  =  */(*)  ./''(*)• 

16.  Show  that  the  maximum  and  minimum  ordinates  of  the  curve 

2/  =  2/(^r(^)-{/'^)}2 

correspond  to  the  points  of  intersection  of  the  curve  y2  =f(z)  with  the  axis 
of  a\ 

17.  When  y2  =f(%)  represents  a  cubic,  prove  that  the  biquadratic  in  x 
which  determines  its  points  of  inflexion  has  one,  and  but  one,  pair  of  real  roots. 
Prove  also  that  the  lesser  of  these  roots  corresponds  to  no  real  point  of  inflexion, 
while  the  greater  corresponds,  in  general,  to  two. 

1 8.  Prove  that  the  point  of  inflexion  of  the  cubic 

ayz  +  ^bxy2  +  T>cx2y  +  dx%  +  30a;2  =  o 

lies  in  the  right  line  ay  +  bx  =  o,  and  has  for  its  co-ordinates 

■\a2e        _         xabe 
*--  -^-,and2/=  — , 

where  G  is  the  same  as  in  Example  32,  p.  190. 

19.  Find  the  nature  of  the  double  point  of  the  curve 

y2  =  {x  -  2)2  (x  -  5), 

and  show  that  the  curve  has  two  real  points  of  inflexion,  and  that  they  subtend 
a  right  angle  at  the  double  point. 

20.  The  co-ordinates  of  a  point  on  a  curve  are  given  in  terms  of  an  angle  6 
by  the  equations 

x  =  sec3  6,    y  =  tan  9  sec2  6 ; 

prove  that  there  are  two  finite  points  of  inflexion  on  the  curve,  and  find  the 
values  of  6  at  these  points. 


(     285     ) 


CHAPTEE  XVII. 

RADIUS  OF  CURVATURE.      E VOLUTES.      CONTACT.      RADII  OF 
CURVATURE  AT  A  DOUBLE  POINT. 

225.  Curvature.  Angle  of  Contingence. — Every  con- 
tinuous curve  is  regarded  as  having  a  determinate  curvature 
at  each  point,  this  curvature  being  greater  or  less  according 
as  the  curve  deviates  more  or  less  rapidly  from  the  tangent  at 
the  point. 

The  total  curvature  of  an  arc  of  a  plane  curve  is  measured 
by  the  angle  through  which  it  is  bent  between  its  extremities — 
that  is,  by  the  external  angle  between  the  tangents  at  these 
points,  assuming  that  the  arc  in  question  has  no  point  of  in- 
flexion on  it.  This  angle  is  called  the  angle  of  contingence  pf 
the  arc. 

The  curvature  of  a  circle  is  evidently  the  same  at  each  of 
its  points. 

To  compare  the  curvatures  of 
different  circles,  let  the  arcs  AB 
and  ab  of  two  circles  be  of  equal 
length,  then  the  total  curvatures 
of  these  arcs  are  measured  by  the 
angles  between  their  tangents,  or 
by  the  angles  ACB  and  acb  at  F. 

their  centres :  but  lg*  3°* 

.-,_  ,     aroAB  arc«5        1      1 

LACB:  Lacb  =  — 777-:-- =  -t^:""' 

AC        ac        AC  ac 

Consequently,  the  curvatures  of  the  two  circles  are  to  each 
other  inversely  as  their  radii;  or  the  curvature  of  a  circle 
varies  inversely  as  its  radius. 

Also  if  As  represent  any  arc  of  a  circle  of  radius  r,  and 
A0  the  angle  between  the  tangents  at  its  extremities,  we  have 

As 
r  =  -— . 


286  Radius  of  Curvature. 

The  curvature  of  a  curve  at  any  point  is  found  by  deter- 
mining the  circle  which  has  the  same  curvature  as  that  of  an 
indefinitely  small  elementary  arc  of  the  curve  taken  at  the 
point. 

226.  Radius  of  Curvature. — Let  ds  denote  an  infi- 
nitely small  element  of  a  curve  at  a  point,  d(p  the  corresponding 

ds 
angle  of  contingence  expressed  in  circular  measure,  then  — 

d<h 

evidently  represents  the  radius  of  the  circle  which  has  the 

same  curvature  as  that  of  the  given  curve  at  the  point. 

This  radius  is  called  the  radius  of  curvature  for  the  point, 
and  is  usually  denoted  by  the  letter  p. 

To  find  an  expression  for  p,  let  the  curve  be  referred  to 
rectangular  axes,  and  suppose  x  and  y  to  be  the  co-ordinates 
of  the  point  in  question ;  then  if  0  denote^the  angle  which  the 
tangent  makes  with  the  axis  of  x,  we  have 

dy  u  d.  tan  $  _  d2y 

^     dx '  dx  dx* 

,   dd>      d2y 
or  sec2^-^-  =  -t^2. 

dx      dx7, 

d6     dd>  dx  d<h  ,    d?y 

Hence  „■'      Sjft-V       W.  (,) 

r      d(f>      dy  d2y 


ds       dx*  dx7. 

At  a  point  of  inflexion  -~  =  o  :  accordingly  the  radius  of 

ax 

curvature  at  such  a  point  is  infinite  :  this  is  otherwise  evident 
since  the  tangent  in  this  case  meets  the  curve  in  three  conse- 
cutive points.     (Art.  222.)  ^ 

Again,  as  the  expression  f  1  +  \-j-\  )    has  always  two 
values,  the  one  positive  and  the  other  negative,  while  the 


Expressions  for  Radius  of  Curvature.  287 

curve  can  have  in  general  but  one  definite  circle  of  curvature 

at  any  point,  it  is  necessary  to  agree  upon  which  sign  is  to  be 

taken.     We  shall  adopt  the  positive  sign,  and  regard  p  as 

d  fj 
being  positive  when  — \  is  positive ;   i.  e.  when  the  curve  is 

ax 

convex  at  the  point  with  respect  to  the  axis  of  x. 

227.  Other  Expressions  for  p. — It  is  easy  to  obtain 
other  forms  of  expression  for  the  radius  of  curvature  ;  thus 
by  Art.  178  we  have 

dx       .  dy 

Hence,  if  the  arc  be  regarded  as  the  independent  variable,  we 
get 

d<b      d2x  dd>      d2y 

-  sin  6  -f  =  -=— .      cos  6  -f  =  -7^, 

y  ds      ds*'  r  ds      ds2> 

from  which,  if  we  squaf e  and  add,  we  obtain 


P 


»  \     /Vo     I  \   /Vo*    /  \    Wo'    /  V      / 


ds  J       \ds2  J       \ds 


Again,  the  equations    dx  =  cos  (pds,     dy  =  sin  <pds, 

ds 
give  by  differentiation  (substituting  —  for  d<j>), 

P 

(ds)2  (dsY 

d2x  =  cos  (pd2s  -  sin  $- — — ,     d2y  =  sin  $d2s  +  cos  0  - — -.      (3) 

Whence,  squaring  and  adding,  we  obtain 

(d2x)2  +  (d2y)2  =  (d2s)2  +  @£, 

P 

ds2 
9     </(d2x)2  +  {d2yf  -  (d2sf  *4' 


288  Radius  of  Curvature. 

Again,  if  the  former  equation  in  (3)  be  multiplied  by- 
sin  0,  and  the  latter  by  cos  0,  we  obtain  on  subtraction, 

d$2  ds^ 

cos  <pd2y  -  sin  $d2x  =  — ,     or  dxd2y  -  dyd2x  =  — . 

.p.  (dec2  +  dy2)% 

^     dxd2y  -  dyd2x 

The  independent  variable  is  undetermined  in  formulae  (4) 
and  (5),  and  may  be  any  quantity  of  which  both  x  and  y  are 
functions. 

For  example,  in  the  motion  of  a  particle  along  a  curve, 
when  the  time  is  taken  as  the  independent  variable,  we  get 
from  (4)  an  important  result  in  Mechanics. 


Examples. 
1.  To  find  the  radius  of  curvature  at  any  point  on  the  parabola  x2  =  qmy. 

„  dy  d2y  /dy\2  x2  y 

Here  2m-f-  =  x,     2m-^  =  i,     i  +  i-f)    =1+ — -=i+—; 

dx  dx1  \dxf  q.m*  m 

2  (m  +  y)* 


.:  p 


vm 


2.  Find  the  radius  of  curvature  in  the  catenary 


Here  —  =  -  (  ea  -  e"a) 

dx      2  \  / 


d2y  _  y     .    _  _  y*_ 


Hence  the  radius  of  curvature  is  equal  to  the  part  of  the  normal  intercepted 
by  the  axis  of  x,  but  measured  in  the  opposite  direction  (Ex.  4,  Art.  171). 

3.  In  the  cubical  parabola  $a2y  =  #3,  we  have 

*%-*  *%-»»('+ ten1- g-^i  ■■■p={ai**i)i- 

dx  dx2  {  \dx)   )  an  2a4z 


General  Expression  for  Radius  of  Curvature.  289 

X2         I/2 

4.  To  find  the  radius  of  curvature  in  the  ellipse  -z  +  —  =  1. 

*      a2      ft 

Let  x  =  a  cos  <p,  then  y  =  b  sin  <£>,  and  \ve  have 

<£c  =  -  «  sin  <pd<p>    d2x  =  —  a  cos  <pd(p2  —  a  sin  ^d2(p, 
dy  =  b  coB<pd<p,        d2y  =  —  b  sin  <£(?02  +  5  cos  ^^. 
Hence  by  formula  (5)  we  obtain 

(a2sin20  +  £3cos2<£)t 


P  = 


«5 


5.  In  the  hypocycloid  #t  +  y§  =  a$,  let  x  —  a  cos3^,  then  y  —  a  sin3</>,  and  re- 
garding <p  as  the  independent  variable,  we  have 

dx  —  —  2>a  cos2<£  sin  <j>  d<p,  d?x  =  3«  cos  <p  d<f>2  (2  sin2  <p  —  cos2  <£), 

<?y  =  3«  sin2^)  cos  <£  d<f>,  d2y  =  $a  sin  <p  dtp2  (2  cos2  ^>  —  sin2  (p), 

whence 
(<fc2  +  dy2)'  =  3«  sin^>  cos  <pd<p,  and  rf!a;^2«/  -  dyd2x  =  -  9a2  sin20  cosPcpdcp3, 

from  which  we  obtain 

p  =  -  3  (ory)*. 

6.  Find  the  radius  of  curvature  at  any  point  of  the  curve 


1 
ea  =  sec 


{I)'  Ans.p-asecfy 


228.  General  Expression  for  Radius  of  Curva- 
ture.— The  value  of  p  becomes  usually  difficult  of  determi- 
nation from  formula  ( 1 )  whenever  y  is  not  given  explicitly  in 
terms  of  x}  that  is,  when  the  equation  of  the  curve  is  of  the 
form 

«  =/(*»  y)  =  °- 

We  proceed  to  show  how  the  equation  for  p  is  to  be  trans- 
formed in  this  case.     Suppose 

du  du  d2u       .      d2u   __  p     d2u      ~ 

dx        '     dy         '     dx2         '    (&?%         '     ^y2 

then,  by  Art.  ioo,  we  have 

tr 


2  go  Radius  of  Curvature, 

Again,  differentiating  this  equation  with  respect  to  x, 
regarding  y  as  a  function  of  x  in  consequence  of  the  given 
equation,  and  observing  that 

rf  dL     dLdy       d  dM     dM dy 

dx  dx       dy  dx      dx       '       dx       dy  dx* 

we  obtain 

dL     dLdy      (dM     dMdy\dy  d*y 

dx      dy  dx      \  dx       dy  dx)  dx  dx2        ' 

A  +  2B%+0%  +  XpL.Oi  (6) 

dx         dx2  dx2  ' 

whence,  on  substituting  -  —  for  — ,  we  obtain 


Consequently 


d2y        AM*-  2BLM+  CL2 

dj?  "  JSP 


{L2  +  M2f  . 

"  ±  AW-  2BLM  +  CL2'  {7) 


Or,  on  replacing  L,  M,  A,  B,  C,  by  their  values, 

MduS*     fdu\2^ 

=  +  )  W  +  \dy) 

P      ~  d2u  fdu\2         d'lu  du  du      d2u  fdu\2 


dx2  \dy)         dxdy  dx  dy      dy2  \dx 

The  result  in  (6)  enables  us  to  determine  the  second  diffe- 
rential coefficient  of  an  implicit  function  in  general;  a  process 
which  is  sometimes  required  in  analysis. 

229.  The  Centre  of  Curvature  is  the  point  of 
intersection  of  two  Consecutive  Normals. — We  shall 
next  proceed  to  consider  the  subject  from  a  geometrical 
point  of  view. 

As  a  circle  which  passes  through  two  infinitely  near 
points  on  a  curve  is  said  to  have  contact  of  the  first  order  with 


Newton's  Method  of  Investigating  Curvature.         291 

the  curve,  so  the  circle  which  passes  through  three  infinitely 
near  points  on  a  curve  is  said  to  have  contact  of  the  second 
order  with  it,  and  is  called  the  circle  of  curvature,  or  the 
osculating  circle  at  the  point. 

Again,  the  centre  of  the  circle  which  passes  through 
three  points,  P,  Q,  R,  is  the  intersection  of  the  perpendicu- 
lars drawn  at  the  middle  points  of  PQ  and  QE ;  but  when 
P,  Q,  B  become  infinitely  near  points  on  a  curve,  the  per- 
pendiculars become  normals,  and  the  centre  of  the  circle 
becomes  the  limiting  position  of  the  intersection  of  two  infinitely 
near  normals  to  the  curve.     (Compare  Art.  37,  note.) 

ds 

From  this  it  is  easily  seen  that  we  obtain  —  for  the  length 

of  the  radius  of  the  circle  in  the  limit,  as  before. 

230.  Newton's  Method  of  investigating  Radii  of 
Curvature. — When  the  equation  of  the  curve  is  algebraic 
and  rational  it  is  easy  to  obtain  an 
expression  for  its  radius  of  curvature* 
at  any  point. 

For,  take  the  origin  0  at  the 
point,  and  the  tangent  and  normal 
for  co-ordinate  axes;  let  P  be  a 
point  on  the  curve  near  to  0,  and 
describe  a  circle  through  P  and  0 
touching  the  axis  of  x;  draw  PN 
perpendicular  to  OX  and  produce 
it  to  meet  the  circle  in  Q ;  then  we  have 

ON2  =  PN.NQ. 

Hence,  if  x  and  y  be  the  co-ordinates  of  P,  we  get 

ON* 


NQ  = 


PN 


X" 

y 


But  when  P  is  infinitely  near  to  0,  NQ  becomes  OD,  the 


*  This  method  of  finding  the  radius  of  curvature  is  indicated  by  Newton 
(Principia,  Lib.  I.,  Sect,  i.,  Lemma  xi.),  and  has  been  adopted  in  a  more  or  less 
modified  form  by  many  subsequent  writers. 

TJ  2 


292  Radius  of  Curvature. 

diameter  of  the  circle  of  curvature,  and  if  p  be  its  radius,  we 
have 

2p  =  limit  of  —  when  x  is  infinitely  small. 

Again,  since  the  axis  of  x  is  the  tangent  at  the  origin, 
the  equation  of  the  curve,  by  Art.  208,  is  of  the  form 

lxy  =  c0x2  +  2cxxy  +  c2y2  +  terms  of  the  third  and  higher  degrees 

=  c$?  +  2cxxy  +  c2y2  +u3  +  u±  +  &c.  (9) 

On  dividing  by  y  we  obtain 

o1  =  cQ—  +  2cxx  +  c2y  +  —  +  &c. 

if  if 

Again,  when  x  is  infinitely  small,  —  becomes  2p,   and 

if 

each*  of  the  other  terms  at  the  right-hand  side  becomes  infi- 
nitely small ;  hence 

h 
P  =  — • 

2C0 

Thus,  for  example,  the  radius  of  curvature  at  the  origin  in 
the  curve 

6y  =  2x%  +  3xy  -  ^y7,  +  x* 

■2 

is  -,  the  axes  being  rectangular. 


W3  2/4 

*  We  have  assumed  above  that  the  terms  — ,  — ,  &c,  become  evanescent 

y  y 

along  with  x  ;  this  can  be  readily  established  as  follows : — 
Let  «3  =  axz  +  &x%y  +  yxy2  +  5«/3, 

then  —  =  a—  +  &z2  +  yxy  +  8*/2 ; 

y       y 

x- 
each  of  the  terms  after  the  first  vanishes  with  x.  while  the  first  "becomes  a  —x, 

y 

or  2apx,  which  also  vanishes  with  x,  when  p  is  finite. 

Similar  reasoning  is  applicable  to  the  terms,  — ,  &c. 

y 


Case  of  Oblique  Axes.  293 

From  the  preceding  it  follows  that  when  the  axis  of  x  is 
a  tangent  at  the  origin,  the  length  of  the  radius  of  curvature 
at  that  point  is  independent  of  all  the  coefficients  except 
those  of  y  and  x*. 

231.  Case  of  Oblique  Axes. — If  the  co-ordinate  axes 
be  oblique,  and  intersect  at  an  angle  w,  then  PQ  no  longer 
passes  through  the  centre  of  the  circle  in  the  limit,  but  be- 
comes the  chord  of  the  circle  of  curvature  which  makes  the 
angle  10  with  the  tangent ;  accordingly,  we  have  in  this  case 

ON2     x2 
2p  sin  id  =  -=—  =  — ,  in  the  limit. 
l^Jy       y 

Hence,  in  the  case  of  oblique  axes,  we  have 

|osmw  =  — .  (10) 

2Cq 

If  bi  and  cQ  have  opposite  signs,  p  is  negative  ;  this 
indicates  that  the  centre  of  curvature  lies  below  the  axis  of  x9 
towards  the  negative  side  of  the  axis  of  y. 

The  preceding  results  show  that  the  radius  of  curvature 
at  the  origin  is  the  same  as  that  of  the  parabola,  by  =  c0x2,  at 
the  same  point ;  and  also  that  the  system  of  curves  obtained 
by  varying  all  the  coefficients  in  (9),  except  those  of  y  and 
x2,  have  the  same  osculating  circle,  in  oblique  as  well  as  in 
rectangular  co-ordinates. 

Again,  as  in  Art.  223,  the  osculating  circle,  since  it  meets 
the  curve  in  three  consecutive  points,  cuts  the  curve  at  the 
point,  in  general,  as  well  as  touches  it. 

If  c0  =  o  in  the  equation  of  the  curve,  and  b,,  be  not  zero, 
the  radius  of  curvature  becomes  infinite,  and  the  origin  is  a 
point  of  inflexion.  This  is  also  evident  from  the  form  of  the 
equation,  since  the  axis  of  x  meets  the  curve  in  this  case  in 
three  consecutive  points. 

232.  In  general,  the  equation  of  a  curve  referred  to  any 
rectangular  axes,  when  the  origin  is  on  the  curve,  may  be 
written  in  the  form 

2b0x  +  2bxy  =  c0x2  +  2cvxy  +  c%y2  +  u3  +  &c. 


294  Radius  of  Curvature. 

Here  b0x  +  bYy  =  o  is  the  equation  of  the  tangent  at  the 
origin ;  and  the  length  of  the  perpendicular  PN,  from  the 
point  (x,  y)  on  this  tangent  is 

b0x  +  biy 


<yw  +  K 


Also,  OP2  =  x2  +  tf,  and  OP2  =  2p  .  PJST  in  the  limit. 

Accordingly,  we  have,  when  x  and  y  are  infinitely  small, 

i      2PN  2b0x  +  ibxy 


P       OP2      (3*  +  y*)yb02  +  b 


c0x2  +  2dxy  +  c2y~       % 

"  G*2  +  */2)  </W  +  W     {p%  +  y2)  ^K  +  b* 

(since  the  point  x,  y  is  on  the  curve). 

Again,  the  terms  contained  in   _  3  „,  &c,  become  evanes- 

°  x2  +  y2 

cent  in  the  limit,  as  before  (see  note,  Art.  230). 
Hence  we  have 

e0  +  2d-  +  c2  - 
1       c0ar  +  2cxxy  +  c2y        x        \xy 

But  for  points  infinitely  near  the  origin  we  have 

V         bo 
OqX  +  bxy  =  o,  or  -  =  -  — . 

X  0\ 

y 
Substituting  this  value  instead  of  -  in  the  preceding  equation, 

x 

it  becomes 

i_  _  cj)?  -  2b0blcl  +  c2b2 

p~  (b2  +  b^        '  {II) 

The  student  will  find  no  difficulty  in  showing  the  identity 
of  this  result  with  that  given  in  (7). 


Radius  of  Curvature  in  terms  of  r  and  p. 


295 


233.  Radii   of  Curvature  of  Inverse    Curves.— It 

may  be  convenient  to  state  here  that  if  two  curves  be  inverse 
to  each  other  with  respect  to  any  origin,  their  osculating  circles 
at  two  inverse  points  are  also  inverse  to  each  other  with  respect 
to  the  same  origin. 

This  property  is  evident  geometrically  from  the  con- 
sideration that  a  circle  is  determined  when  three  points  on 
it  are  given. 

Again,  since  the  centres  of  the  two  inverse  circles  are 
in  directum  with  the  origin,  we  can  construct  the  centre  of 
curvature  at  any  point  on  a  curve,  when  that  for  the  cor- 
responding point  on  the  inverse  curve  is  known. 

Also,  if  the  osculating  circle  at  any  point  on  a  curve 
pass  through  the  origin,  the  corresponding  point  is  a  point  of 
inflexion  on  the  inverse  curve. 

We  shall  next  proceed  to  establish  another  expression  for 
the  radius  of  curvature,  which  is  of  extensive  application  in 
curves  referred  to  polar  co-ordinates. 

234.  Radius  of  Curvature  iu  terms  of  r  and  p. — 
Let  PN  and  PC  be  the  tangent 
and  normal  at  any  point  P  on  a 
curve,  P'N'  and  PC  those  at 
the  infinitely  near  point  iy,  then 
C  is  the  centre  of  curvature  cor- 
responding to  the  point  P.  Let 
0  be  the  origin. 

Join  OC,  and  let  OC  =  B, 
OP  =  r,  OP'  =  /,  ON  =  p, 
ON'  =p\  CP=CP,=  p;  then 
we  have 


OC%  =  OP2  +  CP2-20P.CP,  cos  OP(7, 


or 


S2  =  r*  +  p* 


—  2 


pp. 


In  like  manner  we  have 


Subtracting,  we  get 

r 

ry*      /y* 

r'i  _  r3  =  2p  (p'  -  2i),  or  — 


2P 


p  -  p      r  +  r 


296  Radius  of  Curvature. 

Hence  we  have 

dr      p  dr 

This  formula  can  also  be  deduced  immediately  from  Art. 

193  :  thus 

_,  „     dp      dp  ds         dp        dp  dr  ,  dp 

rcos^  =  PJY  =  -j-  =  ——  =  p—  =  p  —  —  =  p  co$\p  —  ; 
da)      ds  do)        as         dr  as  dr 

dp  dr 

dr9  dp' 

235.   Chord   of  Curvature    through   the    Origin. — 

Let  7  denote  half  the  intercept  made  on  the  line  OP  by  the 
circle  of  curvature,  and  we  evidently  have 

y  =  PWlOPN=p^  =  p~p.  (13) 

This  and  the  preceding  formula  are  of  importance  when- 
ever we  can  express  the  equation  of  the  curve  in  terms  of  the 
lines  represented  by  r  and^. 

Their  use  will  be  illustrated  by  the  following  elementary 
examples : — 

Examples. 

1.  To  find  the  radius  of  curvature  at  any  point  on  a  parabola. 
Taking  trie  focus  as  pole,  the  equation  of  the  curve  in  terms  of  r  and  p 
evidently  is  p2  =  zmr. 


—r  dr     pr        /z)z\i      .  dr 

Hence  p  =  r  —  =  —  =     —     ;  also,  7  —p  —  = 

dp      m       \  m  J  dp 


2.  To  find  the  radius  of  curvature  in  an  ellipse. 
Taking  the  centre  as  origin,  the  equation  of  the  curve  is 

pi 

dr       a2bz 
dp       p6 

3.  To  find  the  radius  of  curvature  in  the  Leniniscate. 
Here,  by  Ex.  3,  Art.  190,  we  have  rz  =  a2p ; 

iLT  Co  V* 

. •.  v2  —  =  «2 ;  hence  p  —  —  ;  also,  7  =  -. 
*     dp  H     3r  3 


Evolutes  and  Involutes. 


297 


4.  To  find  the  chord  of  curvature  which  passes  through  the  origin  in  the 
Cardioid 

r  =  a{\  +  cos0). 
In  this  case,  we  have  r3  =  zap2. 

-rr  dr       1 

-Hence  y  =  P  —  =  -r. 

'dp      3 

5.  To  find  the  radius  of  curvature  at  any  point  on  the  curve  rm  =  am  cos  md. 

Here  rmn  =  amp,  by  Art.  190. 

„  am  r2  .  r 

Hence  p  = - = —  ;  also,  7  =   . 

(m  -t  i)rm"1      (m  +  i)p  m+  1 

This  result  furnishes  a  simple  geometrical  method  of  finding  the  centre  of  cur- 
vature in  all  curves  included  under  this  equation. 

236.  To  prove  that  p  =p  +  -±      If  p  and  w  have  the 

same  signification  as  in  Art.  192,  the  formula  of  that  Art. 
becomes 

ds  d2p 

p=dZ=P+^  ('4) 

Examples. 

1.  In  a  central  ellipse  prove  that 

p  =  *y  a2  cos2  ca  +  b2  sin2a>, 

and  hence  deduce  an  expression  for  the  radius  of  curvature  at  any  point  on  the 
curve. 

2.  In  a  parabola  referred  to  its  focus  as  pole,  prove  that  p  =  m  sec  co,  and 
hence  show  that  p  =  2m  sec3 a. 

237.  Evolutes  and  Involutes. — If  the  centre  of  cur- 
vature for  each  point  on  a  curve  be  ■pl  Vo 
taken,  we  get  a  new  curve  called  the 
evolute  of  the  original  one.  Also,  the 
original  curve,  when  considered  with 
respect  to  its  evolute,  is  called  an  in- 
volute. 

To  investigate  the  connexion  be- 
tween these  curves,  let  Pl9  P2,  -Ps,  &c., 
represent  a  series  of  infinitely  near 
points  on  a  curve;  CX9  C2,  Cz,  &c,  the 
corresponding  centres  of  curvature, 
then  the  lines  P^Ci9  P2C2,  PsC3,  &c, 
are  normals  to  the  curve,  and  the  lines 
OiC2,  C2C39  (73C4,&c.,mayberegardedin 
the  limit  as  consecutive  elements  of  the  evolute ;  also,  since 


"2  p 


2q8  Radius  of  Curvature. 

each  of  the  normals  PXCX,  PzC2,PzCz,&o.,  passes  through  two 
consecutive  points  on  the  evolute,  they  are  tangents  to  that 
curve  in  the  limit. 

Again,  if  px,  p2,  p3,  pi}  &c,  denote  the  lengths  of  the  radii 
of  curvature  at  the  points  Px,  P2,  P3,  Pi,  &c,  we  have 

px  =  PXCX,  p2  =  PA  pz  =  PzCs,  p4  =  P4O4,  &o. ; 
.*.   px  —  p%  =  PX0X  —  x  2C2  =  PiL>x  —  P2C2  =  Cj.G2 ; 

alSO         p%—  pz  =  C2 C3,    pz  —  p±  =  C364,  .  .  .  pn_!  —  pn  =  Cn-XCn  ', 

hence  by  addition  we  have 

P\  ~  Pn  —  GiG2  +  6263  +  C/3C/4  +  .  .  .  +  (7w_i  t7«. 

This  result  still  holds  when  the  number  n  is  increased 
indefinitely,  and  we  infer  that  the  length  of  any  are  of  the 
evolute  is  equal,  in  general,  to  the  difference  between  the  radii  of 
curvature  at  its  extremities. 

It  is  evident  that  the  curve  may  be  generated  from  its 
evolute  by  the  motion  of  the  extremity  of  a  stretched  thread, 
supposed  to  be  wound  round  the  evolute  and  afterwards 
unrolled ;  in  this  case  each  point  on  the  string  will  describe 
a  different  involute  of  the  curve. 

The  names  evolute  and  involute  are  given  in  consequence 
of  the  preceding  property. 

It  follows,  also,  that  while  a  curve  has  but  one  evolute,  it 
can  have  an  infinite  number  of  involutes  ;  for  we  may  regard 
each  point  on  the  stretched  string  as  generating  a  separate 
involute. 

The  curves  described  by  two  different  points  on  the 
moving  line  are  said  to  be  parallel;  each  being  got  from  the 
other  by  cutting  off  a  constant  length  on  its  normal  measured 
from  the  curve. 

238.  E volutes  regarded  as  Envelopes. — From  the 
preceding  it  also  follows  that  the  determination  of  the  evolute 
of  a  curve  is  the  same  as  the  finding  the  envelope  of  all  its 
normals.  "We  have  already,  in  Ex.  3,  Art.  219,  investigated 
the  equation  of  the  evolute  of  an  ellipse  from  this  point  of 
view. 

239.  Evolute  of  a  Parabola. — "We  proceed  to  deter- 
mine the  evolute  of  the  parabola  in  the  same  manner. 


Evolute  of  Ellipse. 


299 


Let  the  equation  of  the  curve  be  y2  =  2 ma?,  then  that  of 
its  normal  at  a  point  (%,  y)  is 


m 


or 


[Y-y)-  +  X-x  =  o, 

is 

yz  +  2my  (m  -  X)  -  2m2 Y  =  o. 


The  envelope  of  this  line,  where  y  is  regarded  as  an  arbi- 
trary parameter,  is  got  by  eliminating  y  between  this  equa- 
tion and  its  derived  equation 

$y2  +  2m  (m  -  X)  =  o. 

Accordingly,  the  equation  of  the 
required    envelope    is   obtained   by 

Y 

instead  of  y 


substituting 

2   m 


X 


Fig.  34. 


in  the  latter  equation. 

Hence,  we  get  for  the  required 
evolute,  the  semi-cubical  parabola 

2jmY2  =  8  ( X  -  m)3. 

The  form  of  this  evolute  is  exhi- 
bited in  the  annexed  figure,  where 
VN=m  =  2VF.  ^  If  P,  P',  repre- 
sent the  points  of  intersection  of  the 
evolute  with  the  curve,  it  is  easily  seen  that 

VM  =  4VJST  =  4m. 

240.  Evolute  of  an  Ellipse. — The  form  of  the  evolute  of 
an  ellipse,  when  e  is  greater 

than  -J-v  2,  is  exhibited  in 
the  accompanying  figure ; 
the  points  M,  iV,  H ',  N',  are 
evidently  cusps  on  the  curve, 
and  are  the  centres  of  cur- 
vature corresponding  to  the 
four  vertices  of  the  ellipse. 
In  general,  if  a  curve  be 
symmetrical  at  both  sides 
of  a  point  on  it,  the  oscu- 
lating circle  cannot  intersect 


3°°  Radius  of  Curvature. 

the  curve  at  the  point ;  accordingly,  the  radius  of  curvature 
is  a  maximum  or  a  minimum  at  such  a  point,  and  the  corre- 
sponding point  on  the  evolute  is  a  cusp. 

It  can  be  easily  seen  geometrically  that  through  any  point 
four  real  normals,  or  only  two,  can  be  drawn  to  an  ellipse, 
according  as  the  point  is  inside  or  outside  the  evolute. 

It  may  be  here  observed  that  to  a  point  of  inflexion  on 
any  curve  corresponds  plainly  an  asymptote  to  its  evolute. 

241.  Evolute  of  an  Equiangular  Spiral. — We  shall 
next  consider  the  equiangular  or  logarithmic  spiral,  r  =  a0. 

Let  P  and  Q  be  two  points 
on  the  curve,  0  its  pole,  PC, 
QC 'the  normals  at  P  and  Q;  join 
OC.  Then  by  the  fundamental 
property  of  the  curve  (Art.  181), 
the  angles  OP C  and  OQC  are 
equal,  and  consequently  the  four 
points,  0,  P,  Q,  C,  lie  on  a  circle : 
hence  L  QOC  =  L  QPC;  but  in 
the  limit  when  P  and  Q  are  coin-  -p.       6 

cident,  the  angle  QPC  becomes 

a  right  angle,  and  C  becomes  the  centre  of  curvature  belong- 
ing to  the  point  P ;  hence  POC  also  becomes  a  right  angle, 
and  the  point  C  is  immediately  determined. 

Again,  L  OCP  =  L  OQP ;  but,  in  the  limit,  the  angle 
OQP  is  constant;  .*.  L  OCP  is  also  constant ;  and  since  the 
line  CP  is  a  tangent  to  the  evolute  at  C,  it  follows  that  the 
tangent  makes  a  constant  angle  with  the  radius  vector  OC. 
From  this  property  it  follows  that  the  evolute  in  question  is 
another  logarithmic  spiral.  Again,  as  the  constant  angle  is 
the  same  for  the  curve  and  for  its  evolute,  it  follows  that  the 
latter  curve  is  the  same  spiral  turned  round  through  a  known 

angle  (whose  circular  measure  is loga  M) . 

241  (a).  Involute  of  a  Circle. — As  an  example  of 
involutes,  suppose  APQ  to  represent  a  portion  of  an  involute 
of  the  circle  BA  C,  whose  centre  is  0.     Let 

OC  =  a,    L  CO  A  =  0, 

and  CA  the  length  of  the  string  unrolled ;  then 

CP=  CA  =  a<t>. 


Points  of  Inflexion  in  Polar  Co-ordinates. 


301 


Draw  ON  perpendicular  to  the  tangent  at  P,  and  let 
ON  =  p,  then  we  have  *?  •*.  +  \ 

p  =  a$.        D^ 

Hence,  since 

z£Oi\r=zC<X4  =  0, 

the  pedal  of  the  curve  APQ  is  a 
spiral  of  Archimedes. 
Also,  since 

OP2  =  OC2  +  CP\ 
we  have 


r%  _  p2  +  ^ 


Fig-  37- 


which  gives  the  equation  to  the  involute  of  a  circle  in  terms 
of  the  co-ordinates  r  and  p. 
Again,  if  AP  =  s,  we  have 


ds 
d(p 


=  CP  =  acj>; 


from  which  it  is  easily  seen  that 

aft 


s  = 


242.  Radius  of  Curvature,  and  Points  of  In- 
flexion, in  Polar  Co-ordinates. — "We  shall  first  find  an 
expression  for  p  in  terms  of  u  (the  reciprocal  of  the  radius 
vector)  and  0. 

By  Article  1 8$  we  have 


1        _     fdu\" 


hence 
Also 


P 


1  dp  dhc 

p3du  d6z' 


dr 


1  du 


P  =  rT~  =  "17' 
dp        w  dp 


302  Radius  of  Curvature. 

consequently    P  (u  +  ^-2J  = —3  =  j  i  +  {-^     - 

fdu\2^ 
1  +  \u~dOJ 


(15) 


i         ,         du         i  dr 
Again,  since  u  =  -,  we  have  ^  =  -^  ^, 

rf2«  _  2  /<&A2    I  rfV 
and  rffl5  =  r3\rf9j~r25^; 

p"- — rfv — 73*'  (l6) 

This  result  can  also  be  established  in  another  manner,  as 
follows: — 

On  reference  to  the  figure  of  Art.  1 8o,  it  is  obvious  that 
Q  =  0  +  \p ;  where  </>  is  the  angle  the  tangent  at  P  makes  with 
the  prime  vector  OX. 

d6  d\p  dd>  ds  d\L 

Hence  dO=I+T9>    °Tds-dd  =  I+T9> 

dip 

i      dtp  dO 

p      ds  ds 

dd 

dr  d  r 

Again,   denoting   -^  and  -^  by  /   and  r",    we   have 


r 
tan  \b  =  — ;  and  hence 


'2  MnJf  Mf1  ,v,Jf 


dd>          ,  ,  r2-rr  r2-n 

dd                    r  f*  +  r* 

1+f  ^-rf^r\  ^ds 

dd            r2  +  r2  dO 


Intrinsic  Equation  of  a  Curve.  303 


9  ff  /9  * 

r  -  rr    +  zr 


Hence,  we  get        p  = 

Or,  replacing  /  and  /'  by  their  values, 


r%+i% 


2\i 


d2r        fdr\2 


Again,  since  p  =  00  at  a  point  of  inflexion,  we  infer  that 
the  points  of  intersection  of  the  curve  represented  by  the 
equation 


^-rw+2W  ~ 

with  the  original  curve,  determine  in  general  its  points  of 
inflexion. 

In  some  cases  the  points  of  inflexion  can  be  easier  found 
by  aid  of  (15),  which  gives,  when  p  =  00, 

dzu 

Examples. 
1.  Find  the  radius  of  curyature  at  any  point  in  the  spiral  of  Archimedes, 

(1  +  e»)i 


r  =  ad.  Ans.  a 


2  +  02 


2.  Find  the  radius  of  curvature  of  the  logarithmic  spiral  r  =  a  . 

Ans.  r  (1  +  (loga)2)=. 

3.  Find  the  points  of  inflexion  on  the  curve 

9 
r  =  29  —  11  cos  26.  Ans.  cos  20  =  — . 

1 1 

4.  Prove  that  the  circle  r  =  10  intersects  the  curve 

r  =  11  —  2  cos  50 

in  its  points  of  inflexion. 

5.  Prove  that  the  curve 

r  =  a  +  b  cos  nd 

has  no  real  points  of  inflexion  unless  a  is  >  b  and  <  (1  +  w2)  b.  "When  a  lies  be- 
tween these  limits,  prove  that  all  the  points  of  inflexion  lie  on  a  circle ;  and  show 
how  to  determine  the  radius  of  the  circle. 


304  Radius  of  Curvature. 

2\2  (a).  Intrinsic  Equation  of  a  Curve. — In  many 
cases  the  equation  of  a  curve  is  most  simply  expressed  in 
terms  of  the  length,  s,  of  the  curve,  measured  from  a  fixed 
point  on  it,  and  the  angle,  0,  through  which  it  is  bent, 
i.  e.  the  angle  of  deviation  of  the  tangent  at  any  point  from 
the  tangent  at  the  fixed  point,  taken  as  origin.  These  are 
styled  the  intrinsic  elements  of  the  curve  by  Dr.  Whewell,* 
to  whom  this  method  of  discussing  curves  is  due. 

The  relation  between  the  length  s  and  the  deviation  <j>  for 
any  curve  is  called  its  intrinsic  equation. 

If  this  relation  be  represented  by  the  equation 

•  -/(*). 

then  if  p  be  the  radius  of  curvature  at  any  point,  we  have 

»  =  |  =/'<*>• 

Also,  if  sy  denote  the  length  of  the  evolute,  from  Art.  237 
it  is  easily  seen  that  the  equation  of  the  evolute  is  of  the  form 

Si  =f'(<p)  +  const. 

From  this  it  follows  that  the  series  of  successive  evolutes 
are  in  this  case  easily  determined  by  successive  differentiation. 

The  simplest  case  of  an  intrinsic  equation  is  that  of  the 
circle,  in  which  case  we  have 

s  =  ad>. 

Again,  from  Art.  241(a),  the  intrinsic  equation  of  the 
involute  of  a  circle  is  reducible  to  the  form 

ad)2 
s  =  — — . 
2 

We  shall  meet  with  further  examples  of  intrinsic  equa- 
tions subsequently. 

243.  Contact  of  Different  Orders. — As  already 
stated,  the  tangent  to  a  curve  has  a  contact  of  the  first  order 
with  the  curve  at  its  point  of  contact,  and  the  osculating 
circle  a  contact  of  the  second  order.  We  now  proceed  to 
distinguish  more  fully  the  different  orders  of  contact  between 
two  curves. 

*  Cambridge  Philosophical  Transactions,  Vols.  viii.  and  ix. 


Contact  of  Different  Orders.  305 

Suppose  the  curves  to  be  represented  by  the  equations 

y=f(x),  and  y  =  0(a?), 

and  that  xx  is  the  abscissa  of  a  point  common  to  both  curves, 
then  we  have 

f(xx)  -#'(*i). 

Again,  substituting  xx  +  h,  instead  of  x  in  both  equations, 
and  supposing  yx  and  y%  the  corresponding  ordinates  of  the 
two  curves,  we  have 

7  2 

yz  =  <j>  (xx  +  h)  =  (j>  (xi)  +  h$'(xx)  + $"(^0  +  &c- 

Subtracting,  we  get 

tfi  -  v> = * l/fa)  -  *' W )  +  ^  IT  M  -  #"(*0 )  +  &o.     (17) 

Now,  suppose  /'(#i)  =  0'(^O>  or  that  the  curves  have  a 
common  tangent  at  the  point,  then 

*  -  *  -  r-,  {/'W  -  ♦*  W )  +  ^rr  (TW  -  *">.) }+  &o. 

1.2  1.2.3 

In  this  case  the  curves  have  a  contact  of  the  first  order ; 
and  when  h  is  small,  the  difference  between  the  ordinates  is 
a  small  quantity  of  the  second  order,  and  as  yx  -  y2  does  not 
change  sign  with  h,  the  curves  do  not  cross  each  other  at  the 
point. 

If,  in  addition 

/»(«)  =  *>>), 

then        y,  -  y,  = {/">0  -  f"(*i)  I  +  &o. 

1.2.3 

In  this  case  the  difference  between  the  ordinates  is  an  in- 
finitely small  magnitude  of  the  third  order  when  h  is  taken 
an  infinitely  small  magnitude  of  the  first;  the  curves  are 
then  said  to  have  a  contact  of  the  second  order ',  and  approach 
infinitely  nearer  to  each  other  at  the  point  of  contact  than  in 

v 


306  Radius  of  Curvature. 

the  former  case.  Moreover,  since  yx  -  y2  changes  its  sign 
with  h,  the  curves  cut  each  other  at  the  point  as  well  as  touch. 
If  we  have  in  addition  f"(x\)  =  0'"(#i),  the  curves  are 
said  to  have  a  contact  of  the  third  order :  and,  in  general,  if 
all  the  derived  functions,  up  to  the  nth  inclusive,  be  the  same 
for  both  curves  when  x  =  xX}  the  curves  have  a  contact  of  the 
nth  or(jerj  an(j  we  have 

Vl  -  y,  =  ^—  {/(»«)  (*,)  -  ^)  («0  j  +  &e.  (i 8) 


n  +  i 


Also,  if  the  contact  be  of  an  even  order,  n  +  i  is  odd,  and 
consequently  hn+l  changes  its  sign  with  h,  and  hence  the  curves 
cut  eac  other  at  their  point  of  contact ;  for  whichever  is  the 
lower  at  one  side  of  the  point  becomes  the  upper  at  the 
other  side. 

If  the  curves  have  a  contact  of  an  odd  order,  they  do  not 
cut  each  other  at  their  point  of  contact. 

From  the  preceding  discussion  the  following  results  are 
immediately  deduced : — 

(i) .  If  two  curves  have  a  contact  of  the  nth  order,  no  curve 
having  with  either  of  them  a  contact  of  a  lower  order  can 
fall  between  the  curves  near  their  point  of  contact. 

(2).  Two  curves  which  have  a  contact  of  the  nth  order  at 
a  point  are  infinitely  closer  to  one  another  near  that  point 
than  two  curves  having  a  contact  of  an  order  lower  than 
the  nth. 

(3).  If  any  number  of  curves  have  a  contact  of  the  second 
order  at  a  point,  they  have  the  same  osculating  circle  at  the 
point. 

244.  Application  to  Circle. — It  can  be  easily  verified 
that  the  circle  which  has  a  contact  of  the  second  order  with  a 
curve  at  a  point  is  the  same  as  the  osculating  circle  determined 
by  the  former  method. 

For,  let  (X-a)2  +  (r-/3)2  =  i22 

be  the  equation  of  a  circle  having  contact  of  the  second  order 
at  the  point  (x,  y)  with  a  given  curve ;  then,  by  the  preceding, 

the  values  of  —  and  -7-f  must  be  the  same  for  the  circle  and 
ax  dx~ 

for  the  curve  at  the  point  in  question. 


Application  to  Circle. 


307 


Differentiating  the  equation  of  the  circle  twice,  and  sub- 
stituting x  and  y  for  X  and  T,  we  get 


and 


x  -  a  +  {y  -  /3)  ^  =  o, 


\dx 


1  + 


Hence    y  -  )3  =  - 


\dx 


<py 
dx2 


x  -  a 


dy 
dx 


1  + 


ay 
dx 


.-.  R2  =  {x-  a)2  +  (y-  /32)  = 


dy 
dx2 

\dx 


(19) 
(20) 

(21) 


(Py\ 
dx2 


This  agrees  with  the  expression  for  the  radius  of  curvature 
found  in  Art.  226. 

The  co-ordinates  a,  j3  of  the  centre  of  curvature  can 
be  found  by  aid  of  equations  (21) ;  and  the  equation  of  the 
e  olute  by  the  elimination  of  x  and  y  between  these  equa- 
tions and  that  of  the  curve. 

In  practice,  the  following  equations  are  often  more  useful : 
thus,  by  differentiation  with  respect  to  x,  we  get  from  (19), 


cPy 

dx~ 


•    j  ji  j~.  \  J 


dx 


dy 

dx 


(22) 


In  like  manner,  from  the  equation 


we  obtain 


(y-(3)+  (x-a)-^  =  o, 


d2x  d  f   dx 

a  dy2  dy\  ~dy 


(23) 


245.  Centre  of  Curvature,  and  Evolute  of  Ellipse. 

-As  an  illustration,  we  shall  apply  these  equations  to  de- 

x  2 


308  Radius  of  Curvature. 

termine  the  co-ordinates  of  the  centre  of  curvature,  and  the 
equation  of  the  evolute  of  the  ellipse 

x2     y2  _ 
a2  +  Y2=l' 

_  dy        b2        dx        a% 

Here  y-f-  =  --^,  a?— =  -T%y\ 

* dx        a2        dy        b2 

d  (   dy\        b2       d  (   dx\        a2 
dx\   dx)        a2'     dy  \  dy 


HenCe  ydtf~     a2     \dx)~     a2     aUf 


b2  (        b2x2 
=  --:    i  + 


a2y2)         a2y2' 


In  like  manner,  we  have 

d2x  a* 


x 


dy2        b2x2 


Substituting  in  (22)  and  (23),  we  obtain  for  the  co-ordinates 
of  the  centre  of  curvature 

_         {a2-b2)y*  (a2-b2)x* 

Again,  substituting  the  values  of  x  and  y  given  by  these 

X"       v 

equations,  in  the  equation  -  +  ^=i,we  get  for  the  equation 

a       0 

of  the  evolute 

(aa)l  +  (j36)l  =  (a2  -  b2)h 

246.  It  may  be  noticed  that  the  osculating  circle  cuts  the 
curve  in  general,  as  well  as  touches  it.  This  follows  from 
Article  243,  since  the  circle  has  a  contact  of  the  second  order 
at  the  point. 

At  the  points  of  maximum  and  minimum  curvature  the 


Osculating  Curves.  309 

osculating  circle  has  a  contact  of  the  third  order  with  the 
curve  ;  for  example,  at  any  of  the  four  vertices  of  an  ellipse 
the  osculating  circle  has  a  contact  of  the  third  order,  and  does 
not  cut  the  curve  at  its  point  of  contact  (Art.  240). 

247.  Osculating  Curves. — "When  the  equation  of  a 
curve  contains  a  numher,  n,  of  arbitrary  coefficients,  we  can 
in  general  determine  their  values  so  that  the  curve  shall  have 
a  contact  of  the  (n  -  i)th  order  with  a  given  curve  at  a  given 
point ;  for  the  n  arbitrary  constants  can  be  determined  so 
that  the  n  quantities 

dy     cPy  dn~ly 

P'  dx'     dx2'  '  '  'oWv 

shall  be  the  same  at  the  point  in  the  proposed  as  in  the 
given  curve,  and  thus  the  curves  will  have  a  contact  of  the 
(n  -  i)th  order. 

The  curve  thus  determined,  which  has  with  a  given  curve 
a  contact  of  the  highest  possible  order,  is  called  an  osculating 
curve,  as  having  a  closer  contact  than  any  other  curve  of  the 
same  species  at  the  point. 

For  instance,  as  the  equation  of  a  circle  contains  but 
three  arbitrary  constants,  the  osculating  circle  has  a  contact 
of  the  second  order,  and  cannot,  in  general,  have  contact  of  a 
higher  order ;  similarly,  the  osculating  parabola  has  a  contact 
of  the  third  order ;  and,  since  the  general  equation  of  a  conic 
contains  five  arbitrary  constants,  the  general  osculating  conic 
has  a  contact  of  the  fourth  order.  In  general,  if  the  greatest 
number  of  constants  which  determine  a  curve  of  a  given 
species  be  n,  the  osculating  curve  of  that  species  has  a  contact 
of  the  (n  -  i)th  order. 

248.  Geometrical  Method. — The  subject  of  contact 
admits  also  of  being  considered  in  a  geometrical  point  of  view ; 
thus  two  curves  have  a  contact  of  the  first  order,  when  they 
intersect  in  ttvo  consecutive  points  ;  of  the  second,  if  they  inter- 
sect in  three ;  of  the  nth,  if  in  n  +  1 .  For  a  simple  investi- 
gation of  the  subject  in  this  point  of  view  the  student  is 
referred  to  Salmon's  Conic  Sections,  Art.  239. 

249.  Curvature  at  a  Double  Point. — We  now  pro- 
ceed to  consider  the  method  of  finding  the  radii  of  curvature 
of  the  two  branches  of  a  curve  at  a  double  point. 


310  Radius  of  Curvature, 

In  this  case  the  ordinary  formula  (8)  becomes  indetermi- 
nate, since 

du  ,  du 

— -  -  o,  and  —  =  o 
dx  dy 

at  a  double  point.  The  question  admits,  however,  of  being 
treated  in  a  manner  analogous  to  that  already  employed  in 
Art.  230  :  we  commence  with  the  case  of  a  node. 

250.  Radii  of  Curvature  at  a  Wode. — Suppose  the 
origin  transferred  to  the  node,  and  the  tangents  to  the  two 
branches  of  the  curve  taken  as  co-ordinate  axes,  w  represent- 
ing the  angle  between  them. 

By  Art.  2 1  o,  the  equation  of  the  curve  is  in  this  case  of 
the  form 

ihxy  =  ax*  +  j3x2y  +  yxy2  +  By3  +  u±  +  &c. : 

dividing  by  xy  we  obtain 

2  2 

2/1  =  a—  +  Bx  +  yy  +  S—  +  —  +  &c. 
y  x       xy 

Now,  let  pi  and  p2  be  the  radii  of  curvature  at  the  origin 
for  the  branches  of  the  curve  which  touch  the  axes  of  x  and  y, 
respectively ;  then,  by  Art.  231,  we  have 

x  fj 

201  sin  (o  =  — ,  and  2p2  sin  w  =  — ,  in  the  limit. 

r  y  '  x 

Again,  it  can  be  readily  seen,  as  in  the  note  to  Art.  230, 

that  the  terms  in  — ,  &c,  become  evanescent  along  with  x 
xy 

x  1./" 

and  y,  and  accordingly  the  limiting  values  of  —  and  —  can 

y  x 

be  separately  found,  as  in  the  Article  referred  to. 

Hence  we  obtain 

h  h  .     . 

Pi=    —• j      P2  =  jT~ = •  (25) 

asinw  0  sin W 

Also,  if  a  =  o,  we  get  pi  =  00,  and  the  corresponding 
branch  of  the  curve  has  a  point  of  inflexion  at  the  origin. 

Similarly,  if  §  =  o,  p2  =  00. 


Radii  of  Curvature  at  a  Cusp.  3 1 1 

If  a  =  o,  and  8  =  o,  the  origin  is  a  point  of  inflexion  on 
both  branches.  This  appears  also  immediately  from  the 
consideration  that  in  this  case  u3  contains  u%  as  a  factor. 

If  the  equation  of  a  curve  when  the  origin  is  at  a  node 
contain  no  terms  of  the  third  degree,  the  origin  is  a  point  of 
inflexion  on  both  branches.  An  example  of  this  is  seen  in 
the  Lemniscate,  Art.  2 1  o. 


Examples. 

x.  Find  the  radii  of  curvature  at  the  origin  of  the  two  branches  of  the  curve 

ax%  —  2bxy  +  cy3  =  #4  +  y*, 

the  axes  being  rectangular.  Ann.  -  and  -. 

a         c 

2.  Find  the  radii  of  curvature  at  the  origin  in  the  curve 

a  (y2,  —  x~)  —  x3. 

Transforming  the  equation  to  the  internal  and  external  bisectors  of  the  angle 
between  the  axes,  it  becomes 

^axy  */  2  =  {x  -  yf  ; 
hence  the  radii  of  curvature  are  2a  \/  2  and  —  2a  v  2,  respectively. 

251.  Radii  of  Curvature  at  a  Cusp. — The  preceding 
method  fails  when  applied  to  a  cusp,  because  the  angle  w 
vanishes  in  that  case.  It  is  easy,  however,  to  supply  an  in- 
dependent investigation  :  for,  if  we  take  the  tangent  and 
normal  at  the  cusp  for  the  axes  of  x  and  y9  respectively,  the 
equation  of  the  curve,  by  the  method  of  Art.  210,  maybe 
written  in  the  form 

y7,  =  ax3  +  $x~y  +  yxy2  +  Sy*  +  ih  +  &c.  (26) 

Now  in  this,  as  in  every  case,  the  curvature  at  the  origin 
depends  on  the  form  of  the  portion  of  the  curve  indefinitely 
near  to  that  point ;  consequently,  in  investigating  this  form 
we  may  neglect  y2x,  y3,  &c,  in  comparison  with  y1 ;  and  xk 
x3y,  &c,  in  comparison  with  x3. 


312  Radius  of  Curvature. 

Accordingly,  the  curvature  at  the  origin  is  the  same,  in 
general,  as  that  of  the  cubic 

y2  =  a%%  +  (5x2y.  (27) 

Dividing  by  x2,  we  get 

V-  =  ax  +  (3y. 

x 

Hence,   in  immediate  proximity  to    the  origin,    -  be- 

x 

comes  very  small,  i.  e.  y  is  very  small  in  comparison  with  x. 
Accordingly,  the  form  of  the  curve  near  the  origin  is  repre- 
sented by  the  equation 

y2  =  axd. 

From  this  we  infer  that  the  form  of  any  algebraic  curve 
near  a  cusp  is,  in  general,  a  semi- cubical  parabola  (see  Ex.  2, 
Art.  211). 


Again,  since 


we  have,  by  Art.  230, 


x*      x 


P-±*J-; 


from  which  we  see  that  p  vanishes  along  with  x,  and  accord- 
ingly the  radii  of  curvature  are  zero  for  both  branches  at  the 
origin. 

This  result  can  also  be  arrived  at  by  differentiation,  by 
aid  of  formula  ( 1 ) . 

252.  Case  where  the  Coefficient  of  xz  is  wanting. — 
Next,  suppose  that  the  term  containing  x3  disappears,  or 
a  =  o,  then  the  equation  of  the  curve  is  of  the  form 

y1  =  (5x2y  +  yxy2  +  §yz  +  aV  +  &c. ; 

and  proceeding  as  before,  the  curvature  at  the  origin  is  the 
same  as  in  the  curve 

y2  =  fix2y  +  aV.  (28) 


Radii  of  Curvature  at  a  Cusp.  3 1 3 

The  two  branches  of  this  curve  are  determined  by  the 
equation 

y=s&^±-y/3a  +  4a/.  (29) 

The  nature  of  the  origin  depends  on  the  sign  of  |32  +  4a',  and 
the  discussion  involves  three  cases. 

(1).  If  |32  +  4a'  be  positive,  it  is  evident  that  the  curve 
extends  at  both  sides  of  the  origin,  and  that  point  is  a  double 
cusp  (Art.  215(a)). 

On  dividing  equation  (28)  by  y2,  and  substituting  2p  for 

x2 

— ,  we  get  1  =  2j3io  +  4a' p2.  (30) 

y 

The  roots  of  this  quadratic  determine  the  radii  of  curva- 
ture of  the  two  branches  at  the  cusp. 

These  branches  evidently  lie  at  the  same,  or  at  opposite 
sides  of  the  axis  of  x,  according  as  the  radii  of  curvature 
have  the  same  or  opposite  signs :  i.  e.  according  as  a  has  a 
negative  or  positive  sign. 

These  results  also  appear  immediately  from  the  circum- 
stance, that  in  this  case  the  form  of  the  curve  very  near  the 
origin  becomes  that  of  the  two  parabolas  represented  by 
equation  (29). 

(2).  If  j32  +  4a  be  negative,  y  becomes  imaginary,  and  the 
origin  is  a  conjugate  point. 

(3).  If  j32  +  4a  =  o,  the  equation  (30)  becomes  a  perfect 
square  :  we  proceed  to  prove  that  in  this  case  the  origin  is  a 
cusp  of  the  second  species. 

To  investigate  the  form  of  the  curve  near  the  origin,  it  is 
necessary  in  this  case  to  take  into  account  the  terms  of  the 
fifth  degree  in  x  (y  being  regarded  as  of  the  second) :  this  gives 

fr  -  fy  -  ^  +  JT*  +  «V  =  ,W  +  J**  ♦  «V).    (3,) 

It  will  be  observed  that  the  right-hand  side  changes  its 
sign  with  x ;  accordingly  the  origin  is  a  cusp.  Also,  the  cusp 
is  of  the  second  species,  for  the  two  roots  of  the  equation  in  y 
plainly  have  the  same  sign,  viz.,  that  of  j3  ;  and  consequently 
both  branches  of  the  curve  at  the  origin  lie  at  the  same  side 
of  the  axis  of  x. 


3 T  4  Radius  of  Curvature. 

Moreover,  as  equation  (30)  has  equal  roots  in  this  case, 
the  radii  of  curvature  of  the  two  branches  are  equal,  and  the 
branches  have  a  contact  of  the  second  order. 

We  conclude  that  when  the  term  involving  x2  in  equation 
(28)  disappears,  the  origin  is  a  double  cusp,  a  cusp  of  the  second 
species,  or  a  conjugate  point,  according  as  j52  +  4a'  >  =  or  <  o. 

Moreover,  if  a  =  o,  one  root  of  the  quadratic  (30)  is  in- 
finite, and  the  other  is  -7^.    The  origin  in  this  case  is  a  double 

2/3 

cusp,  and  is  also  a  point  of  inflexion  on  one  branch.  Such  a 
point  is  called  a  point  of  oscul-infiexion  by  Cramer. 

If  j3  =  o  in  addition  to  a  =  o,  the  origin  is  a  cusp  of  the 
first  species,  but  having  the  radii  of  curvature  infinite  for  both 
branches. 

It  is  easy  to  see  from  other  considerations  that  the  radii 
of  curvature  at  a  cusp  of  the  first  species  are  always  either 
zero  or  infinite. 

For,  since  the  two  branches  of  the  curve  in  this  case 

u     1] 

turn  their  convexities  in  opposite  directions,  — --  must  have 

ax 

opposite  signs  at  both  sides  of  the  cusp,  and  consequently  it 

must  change  its  sign  at  that  point ;  but  this  can  happen  only 

in  its  passage  through  zero,  or  through  infinity. 

It  should  be  observed  that  the  preceding  discussion  applies 
to  the  case  of  a  curve  referred  to  oblique  axes  of  co-ordinates, 
provided  that  we  substitute  y  instead  of  p  ;  where  7  is  half 
the  chord  intercepted  on  the  axis  of  y  by  the  osculating  circle 
at  the  origin. 

253.  Recapitulation. — The  conclusions  arrived  at  in  the 
two  preceding  Articles  may  be  briefly  stated  as  follows  : — 

(1).  Whenever  the  equation  of  a  curve  can  be  transformed 
into  the  shape  y1  =  ax2,  +  terms  of  the  third  and  higher  degrees, 
the  origin  is  a  cusp  of  the  first  species ;  both  radii  of  curva- 
ture being  zero  at  the  point. 

(2).  When  the  coefficient  of  x2  vanishes,*  the  origin  is 

*  In  this  case,  if  v\  be  the  equation  of  the  tangent  at  the  cusp,  the  equation 
of  the  curve  is  of  the  form 

Vl2  +  V1V2  -f-  v±  +  &c.  =  o. 
This  is  also  evident  from  geometrical  considerations. 


General  Investigation  of  Cusps.  315 

generally  either  a  double  cusp,  a  conjugate  point,  or  a  cusp 
of  the  second  species.  In  the  latter  case  the  two  branches 
of  the  curve  have  the  same  centre  of  curvature,  and  conse- 
quently have  a  contact  of  the  second  order  with  each  other. 

(3).  If  the  lowest  term  in  x  (independent  of  y)  be  of  the 
5th  degree,  the  origin  is  a  point  of  oscul-inflexion. 

If,  however,  the  coefficient  of  x2y  also  vanish,  the  origin 
is  not  only  a  cusp  of  the  first  species,  but  also  a  point  of 
inflexion  on  both  branches  of  the  curve. 

254.  Creneral  Investigation  off  Cusps. — The  pre- 
ceding results  admit  of  being  established  in  a  somewhat  more 
general  manner  as  follows  : — 

By  the  method  already  given,  the  equation  which  deter- 
mines the  form  of  an  algebraic  curve  near  to  a  cusp  may  be 
written  in  the  following  general  shape  : 

y2  =  iAxay  +  JBxb  +  Cx°,  (32) 

where  2Axa  is  the  lowest  term  in  the  coefficient  of  y,  and 
Bxb,  Cxc*  are  the  lowest  terms  independent  of  y. 

By  hypothesis,  a,  b,  c  are  positive  integers,  and  a  >  1,  b  >  2, 
c  >  3  ;  now,  solving  for  y,  we  obtain 

y  =  Axa  ±  </A2x2a  +Bxb  +  Cxc, 

which  represents  two  parabolasf  osculating  the  two  branches 
at  the  origin. 

The  discussion  of  the  preceding  form  for  y  resolves  itself 
into  three  cases,  according  as  2a  is  >  =  or  <  b. 

(1).  Let  2a  =  b  +  h,  then 

b  +  h      b 

y  =  AxT±  x2  </B  +  AFxh  +  Cxc~K 

b 

(a).  If  b  be  odd,  x2  becomes  imaginary  for  negative  values 
of  x,  and  accordingly  the  origin  is  a  cusp  of  the 
first  species  in  this  case. 


*  This  term  is  retained,  as  it  is  necessary  in  the  case  of  a  cusp  of  the  second 
species. 

t  The  word  parabola  is  here  employed  in  its  more  extensive  signification. 


3 1 6  Radius  of  Curvature. 

(j3).  If  b  be  even,  and  B  positive,  y  is  real  for  all  values 
of  x  near  the  origin ;  accordingly  that  point  is  a 
double  cusp. 

(7).  If  b  be  even,  and  B  negative,  the  origin  is  a  conjugate 
point. 

(2).  If  2a  =  b,  we  have 

y  =  Axa  ±  xa  «/{Az  +  B)  +  Occ-\ 

In  this  case,  the  origin  is  either  a  double  cusp,  or  a  conju- 
gate point,  according  as  A%  +  B  is  positive  or  negative. 
Again,  if  A2  +  B  =  o,  we  have 

e-b 

y  =  xa(A  +  x  2    ^/O). 

(a).  If  c  -  b  be  an  odd  number,  the  origin  is  a  cusp  of  the 
second  species. 

(j3) .  If  c  -  b  be  even,  the  origin  is  a  double  cusp  or  a  con- 
jugate point  according  as  C  is  positive  or  negative. 

(3).  2a  <  b,  or  b  =  2a  +  h. 


Here  y  =  Axa  ±  xa  */ A2  +  Bxh  +  Cxc'2a, 

and  the  curve  evidently  extends  at  both  sides  of  the  origin, 
which  accordingly  is  a  double  cusp. 

This  method  of  investigating  curvature  is  capable  of  being 
modified  so  as  to  apply  to  the  case  of  multiple  points  of  a 
higher  order ;  the  discussion,  however,  is  neither  sufficiently 
elementary,  nor  sufficiently  important,  to  be  introduced  here. 

255.  Points  on  E volute  corresponding  to  Cusps  on 
Curve. — In  connexion  with  evolutes  and  involutes,  the  pre- 
ceding results  lead  to  a  few  interesting  conclusions. 

(1).  If  a  curve  has  a  cusp  of  the  first  species,  its  evolute 
in  general  passes  through  the  cusp.  However,  if  in  addition 
the  cusp  be  a  point  of  inflexion,  the  tangent  at  it  is  an  asymp- 
tote to  the  evolute. 

(2).  To  a  cusp  of  the  second  species  corresponds  in  general 
a  point  of  inflexion  on  the  evolute :  in  some  cases  the  point 
of  inflexion  lies  altogether  at  infinity. 

(3).  To  a  double  cusp  corresponds  a  double  tangent  to  the 
evolute. 


Equation  of  Osculating  Conic.  317 

256.  Equation   of  the    Osculating    Conic. — As    an 

additional  illustration  of  the  principles  involved  in  the  pre- 
ceding investigation,  it  is  proposed  to  discuss  the  question  of 
the  conic  which  osculates  an  algebraic  curve  at  a  given  point. 
Transferring  the  origin  to  the  point,  and  taking  the  tangent 
as  axis  of  xf  the  equation  of  the  curve  may  be  written  in  the 
form 

ay  =  x2  +  axxy  +  a2y2  +  b0xz  +  bxx2y  +  b2xy2  +  b3y* 

+  CqX4-  +  dxzy  +  &c.  +  d0x5  +  &c.  (33) 

In  considering  the  form  of  the  curve  near  the  origin,  as  a 
first  approximation  we  may,  as  in  Art.  251,  neglect  xy,  y2,  &c, 
in  comparison  with  y ;  and  x3,  xi,  &c,  in  comparison  with  x2 ; 
thus  the  equation  reduces  to  the  form 

ay  =  x\  (34) 

Hence  the  form  to  which  every  curve  of  finite  curvature 
approximates  in  the  limit  is  that  of  the  common  parabola,  as 
already  seen  in  Art.  231. 

To  proceed  to  the  next  approximation,  we  retain  terms  of 
the  third  order  (remembering  that  when  a?  is  a  very  small 
quantity  of  the  first  order,  y  is  one  of  the  second),  and  the 
equation  becomes 

ay  =  x2  +  axxy  +  b0x*. 

On  substituting  ay  instead  of  x2  in  the  term  b0x3,  the  pre- 
ceding equation  becomes 

ay  =  x2  +  (ax  +  b0a)  xy.  (35) 

This  represents  a  conic  having  contact  of  the  third  order 
with  the  proposed  curve  at  the  origin.  When  ax  +  b0a  =  o,  the 
parabola  ay  =  x2  has  a  contact  of  the  third  order  at  the  origin, 
and  accordingly  so  also  has  the  osculating  circle. 

In  proceeding  to  the  next  and  final  approximation,  we  re- 
tain terms  of  the  fourth  order,  and  we  get 

ay  -  x2  +  a  1  cry  +  a2y2  +  b0x3  +  bxx2y  +  £0a?4.  (36) 


3 1 8  Radius  of  Curvature. 

Moreover,  from  the  preceding  approximation  we  have 

b0axy  =  b0x3  +  b^y  {ax  +  ah0). 

Hence,  we  get  for  the  equation  of  the  conic  having  a 
contact  of  the  closest  kind  with  the  given  curve 

ay  =  x2  +  (a i  +  b0a)  xy  +  [a2  +  a  (5,  -  ax  b0)  +  a2  [cQ  -  b<?) ]  y\     (3  7) 

This  conic,  since  it  has  the  closest  contact  possible  with 
the  given  curve  at  the  origin,  is  the  osculating  conic  (Art.  246) 
for  that  point. 

In  like  manner  the  parabola 

ay  =  x2  +  (ff,  +  b0a)  xy  +  v '-  y\  (38) 

since  it  has  the  closest  contact  possible  for  a  parabola,  is  the 
osculating  parabola  at  the  point. 


Examples.  319 


Examples. 

1.  Prove  that  the  radius  of  curvature  at  the  vertex  of  a  parabola  is  equal  to 
its  semi-latus  rectum. 

2.  Find  the  length  of  the  radius  of  curvature  at  the  origin  in  the  curve 

2/4  +  xz  +  a  (#2  +  y2)  —  a2y.  Ans.  -. 

2 

3.  Find  the  radius  of  curvature  at  the  origin  in  the  curve 

art/  =  by?  -f  cx2y.  Ans.  oo. 

4.  Prove  that  the  locus  of  the  centre  of  a  conic  having  contact  of  the  third 
order  with  a  given  curve  at  a  common  point  is  a  right  line. 

5.  Prove  that  the  locus  of  the  centres  of  equilateral  hyperbolas,  which  have 
contact  of  the  second  order  with  a  given  curve  at  a  fixed  point,  is  a  circle,  whose 
radius  is  half  that  of  the  circle  of  curvature  at  the  point. 

6.  Prove  geometrically  that  the  centre  of  curvature  at  any  point  on  an  ellipse 
is  the  pole  of  the  tangent  at  the  point,  with  respect  to  the  confocal  hyperbola 
which  passes  through  that  point. 

7.  The  locus  of  the  centres  of  ellipses  whose  axes  have  a  given  direction,  and 
which  have  a  contact  of  the  second  order  with  a  given  curve  at  a  common  point, 
is  an  equilateral  hyperbola  passing  through  the  point  ? 

8.  Prove  that  the  locus  of  the  focus  of  a  parabola,  which  has  a  contact  of 
the  second  order  with  a  given  curve  at  a  given  point,  is  a  circle. 

9.  Prove  that  the  radius  of  curvature  of  the  curve  am~l  y  =  xm  at  the  origin  is 

a        .        . 
zero,  -,  or  infinity,  according  as  m  is  <  =  or  >  2  :  m  being  assumed  to  be  greater 
2 

than  unity. 

10.  Two  plane  closed  curves  have  the  same  evolute  :  what  is  the  difference 
between  their  perimeters  ? 

Ans.  2ird,  where  d  is  the  distance  between  the  curves. 

1 1 .  Find  the  radius  of  curvature  at  the  origin  in  the  curve 

3^  =  4#-  i5#2  -3a:3: 

find  also  at  what  points  the  radius  of  curvature  is  infinite. 

12.  Apply  the  principles  of  investigating  maxima  and  minima  to  find  the 
greatest  and  least  distances  of  a  point  from  a  given  curve  ;  and  show  that  the 
problem  is  solved  by  drawing  the  normals  to  the  curve  from  the  given  point. 

(a).  Prove  that  the  distance  is  a  minimum,  if  the  given  point  be  nearer  to 
the  curve  than  the  corresponding  centre  of  curvature,  and  a  maximum  if  it  be 
lurther. 


320  Examples, 

(b).  If  the  given  point  be  on  the  evolute,  show  that  the  solution  arrived  at 
is  neither  a  maximum  nor  a  minimum ;  and  hence  show  that  the  circle  of  curva- 
ture cuts  as  well  as  touches  the  curve  at  its  point  of  contact. 

13.  Find  an  expression  for  the  whole  length  of  the  evolute  of  an  ellipse. 

a3  -  P 

Ans.  4 —  . 

ab 

14.  Find  the  radii  of  curvature  at  the  origin  of  the  two  branches  of  the  curve 

c  a 

xi—-  ax2y  —  axy2  +  a2y2  =  o.  Ans.  a  and  -. 

2  4 

15.  Prove  that  the  evolute  of  the  hypocycloid 

#§  +  y%  =  «l 
is  the  hypocycloid 

(a  +  j8)i  +  (a  -  jB)l  =  Mi. 

16.  Find  the  radius  of  curvature  at  any  point  on  the  curve 

y  +  v  x  (l  —  ®)  —  si11"1  V  x' 

17.  If  the  angle  between  the  radius  vector  and  the  normal  to  a  curve  has  a 
maximum  or  a  minimum  value,  prove  that  7  =  r ;  where  7  is  the  semi-chord  of 
curvature  which  passes  through  the  origin. 

18.  If  the  co-ordinates  of  a  point  on  a  curve  be  given  by  the  equations 

x  =  c  sin  20(i  +  cos  20),        y  =  e  cos  20  (1  -  cos  20), 

find  the  radius  of  curvature  at  the  point.  Ans.  4c  cos  30 

19.  Show  that  the  evolute  of  the  curve 

r2  -  a2  =  mp2 

has  for  its  equation 

r2  —  (1  -  m)  a2  —  mp2. 

20.  If  a  and  j8  be  the  co-ordinates  of  the  point  on  the  evolute  corresponding 
to  the  point  (x,  y)  on  a  curve,  prove  that 

dy     da 
dx     dfi 

21.  If  p  be  the  radius  of  curvature  at  any  point  on  a  curve,  prove  that  the 

radius  of  curvature  at  the  corresponding  point  in  the  evolute  is  -7- ;  where  a> 

aco 

is  the  angle  the  radius  of  curvature  makes  with  a  fixed  line. 

22.  In  a  curve,  prove  that 


1       d   (dy\ 
p      dx  \ds  J  ' 


Examples.  321 

23.  Find  the  equation  of  the  evolute  of  an  ellipse  by  means  of  the  eccentric 
angle. 

24.  Prove  that  the  determination  of  the  equation  of  the  evolute  of  the 
curve  y  =  Jcxn  reduces  to  the  elimination  of  %  between  the  equations 

n  -  2          Fn2     .    .         ,  .      2»  -  1  i 

a  = g a2""1,  and  £  = #*»  + 


n-i        n-i         '  w-i  &«(*»-  i)#M-* 

25.  In  figure,  Art.  239,  if  the  tangent  to  the  evolute  at  P  meet  the  parabola 
in  a  point  2Z,  prove  that  EN  is  perpendicular  to  the  axis  of  the  parabola. 

26.  If  on  the  tangent  at  each  point  on  a  curve  a  constant  length  measured 
from  the  point  of  contact  be  taken,  prove  that  the  normal  to  the  locus  of  the 
points  so  found  passes  through  the  centre  of  curvature  of  the  proposed  curve. 

27.  In  general,  if  through  each  point  of  a  curve  a  line  of  given  length  be 
drawn  making  a  constant  angle  with  the  normal,  the  normal  to  the  curve  locus 
of  the  extremities  of  this  line  passes  through  the  centre  of  curvature  of  the  pro- 
posed.    (Eertrand,  Cal.  Dif.,  p.  573.) 

This  and  the  preceding  theorem  can  be  immediately  established  from  geome- 
trical considerations. 

28.  If  from  the  points  of  a  curve  perpendiculars  be  drawn  to  one  of  its  tan- 
gents, and  through  the  foot  of  each  a  line  be  drawn  in  a  fixed  direction,  pro- 
portional to  the  length  of  the  corresponding  perpendicular ;  the  locus  of  the 
extremity  of  this  line  is  a  curve  touching  the  proposed  at  their  common  point. 
Find  the  ratio  of  the  radii  of  curvature  of  the  curves  at  this  point. 

29.  Find  an  expression  for  the  radius  of  curvature  in  the  curve  p  = 

\/ m2— r2' 
p  being  the  perpendicular  on  the  tangent. 

30.  Being  given  any  curve  and  its  osculating  circle  at  a  point,  prove  that 
the  portion  of  a  parallel  to  their  common  tangent  intercepted  between  the  two 
curves  is  a  small  quantity  of  the  second  order,  when  the  distances  of  the  point 
of  contact  from  the  two  points  of  intersection  are  of  the  first  order. 

Prove  that,  under  the  same  circumstances,  the  intercept  on  a  line  drawn 
parallel  to  the  common  normal  is  a  small  quantity  of  the  third  order. 

31.  In  a  curve  referred  to  polar  co-ordinates,  if  the  origin  be  taken  on  the 

curve,  with  the  tangent  at  the  origin  as  prime  vector,  prove  that  the  radius  of 

r 
curvature  at  the  origin  is  equal  to  one-half  the  value  of  -  in  the  limit. 

d 

32.  Hence  find  the  length  of  the  radius  of  curvature  at  the  origin  in  the 

A  n(i 

curve  r  =  a  sin  nd.  Am.  p  =  — 

2 

33.  Find  the  co-ordinates  of  the  centre  of  curvature  of  the  catenary ;  and 
show  that  the  radius  of  curvature  is  equal,  but  opposite,  to  the  normal. 

34.  If  p,  p'  be  the  radii  of  curvature  of  a  curve  and  of  its  pedal  at  corre- 
sponding points,  show  that 

p'(2r2  —pp)  =r3. 

Ind.  Civ.  Ser.  Exam.y  1878. 

Y 


(       322       ) 


CHAPTER  XVIII. 

ON   TRACING   OF    CURVES. 

257.  Tracing  Algebraic  Curves. — Before  concluding  the 
discussion  of  curves,  it  seems  desirable  to  give  a  brief  state- 
ment of  the  mode  of  tracing  curves  from  their  equations. 

The  usual  method  in  the  case  of  algebraic  curves  consists 
in  assigning  a  series  of  different  values  to  one  of  the  co-ordi- 
nates, and  calculating  the  corresponding  series  of  values  of 
the  other ;  thus  determining  a  definite  number  of  points  on 
the  curve.  By  drawing  a  curve  or  curves  of  continuous  cur- 
vature through  these  points,  we  are  enabled  to  form  a  tolerably 
accurate  idea  of  the  shape  of  the  curve  under  discussion. 

In  curves  of  degrees  beyond  the  second,  the  preceding 
process  generally  involves  the  solution  of  equations  beyond 
the  second  degree  :  in  such  cases  we  can  determine  the  series 
of  points  only  approximately. 

258.  The  following  are  the  principal  circumstances  to  be 
attended  to : — 

(1).  Observe  whether  from  its  equation  the  curve  is  sym- 
metrical with  respect  to  either  axis;  or  whether  it  can  be 
made  so  by  a  transformation  of  axes.  (2).  Find  the  points 
in  which  the  curve  is  met  by  the  co-ordinate  axes.  (3).  De- 
termine the  positions  of  the  asymptotes,  if  any,  and  at  which 
side  of  an  asymptote  the  corresponding  branches  lie.  (4).  De- 
termine the  double  points,  or  multiple  points  of  higher  orders, 
if  any  belong  to  the  curve,  and  find  the  tangents  at  such 
points  by  the  method  of  Art.  212.  (5).  The  existence  of 
ovals  can  be  often  found  by  determining  for  what  values  of 
either  co-ordinate  the  other  becomes  imaginary.  (6).  If  the 
curve  has  a  multiple  point,  its  tracing  is  usually  simplified  by 
taking  that  point  as  origin,  and  transforming  to  polar  co-or- 
dinates :  by  assigning  a  series  of  values  to  6  we  can  usually 
determine  the  corresponding  values  of  r9  &c.     (7).  The  points 


On  Tracing  of  Curves. 


323 


where  the  y  ordinate  is  a  maximum  or  a  minimum  are  found 

du 
from  the  equation  ~  =  o  :  by  this  means  the  limits  of  the 

(XX 

curve  can  be  often  assigned.     (8).  Determine  when  possible 
the  points  of  inflexion  on  the  curve. 

259.  To  trace  the  Curve  y2  =  x2  (x  -  a) ;  a  being  sup- 
posed positive. 

In  this  case  the  origin  is 
a  conjugate  point,  and  the 
curve  cuts  the  axis  of  #  at  a 
distance  OA  =  a.  Again, 
when  x  is  less  than  a,  y  is 
imaginary,  consequently  no 
portion  of  the  curve  lies  to 
the  left-hand  side  of  A. 

The  points  of  inflexion,  I ' 

and  I\  are  easily  determined 

d2y         ,  FiS-  38. 

from  the  equation  -7-f =  o ;  the 

CLX 

corresponding  value  of  x  is  — ;  accordingly  AN . 

Again,  if  TI  be  the  tangent  at  the  point  of  inflexion  7,  it 

a      AN 

can  readily  be  seen  that  TA  =  —  = . 

J  9        3. 

This  curve  has  been  already  considered  in  Art.  213,  and 
is  a  cubical  parabola  having  a  conjugate  point. 

260.  Cubic  with  three  Asymptotes. — We  shall  next 
consider  the  curve* 


y2x  +  ey  =  ax%  +  bx2  +  ex  +  d, 


(1) 


where  a  is  supposed  positive. 

The  axis  of  y  is  an  asymptote  to  the  curve  (Art.  200),  and 
the  directions  of  the  two  other  asymptotes  are  given  by  the 
equation 

y2  -  ax2  =  o,     or  y  =  ±  x  \/a. 


*  This  investigation  is  principally  taken  from  Newton's  Enumeratio  Zi- 
nearum  Tertii  Ordinis. 

Y  2 


324  On  Tracing  of  Curves. 

If  the  term  bxz  be  wanting,  these  lines  are  asymptotes  ;  if  b 
be  not  zero,  we  get  for  the  equation  of  the  asymptotes 

s-        b  ,-        b 

y  =  x*/a  + — — ,       y  +  x*/a  + — —  =  o. 

2\/a  2y  a 

On  multiplying  the  equations  of  the  three  asymptotes 
together,  and  subtracting  the  product  from  the  equation  of 
the  curve,  we  get 

b* 
ey  =  (c )  x  +  d : 

this  is  the  equation  of  the  right  line  which  passes  through  the 
three  points  in  which  the  cubic  meets  its  asymptotes.  (Art. 
204.) 

Again,  if  we  multiply  the  proposed  equation  by  x,  and 
solve  for  xy,  we  get 


xy 


el  62 

=  —  ±    lax4,  +  bxz  +  ex2  +  dx  +  -  :  (2) 

2     \  4 


from  which  a  series  of  points  can  be  determined  on  the  curve 

corresponding  to  any  assigned  series  of  values  for  x. 

It  also  follows  that  all  chords  drawn  parallel  to  the  axis 

e 
of  y  are  bisected  by  the  hyperbola  xy  +  -  =  o  :  hence  we  infer 

that  the  middle  points  of  all  chords  drawn  parallel  to  an 
asymptote  of  the  cubic  lie  on  a  hyperbola. 

The  form  of  the  curve  depends  on  the  roots  of  the  bi- 
quadratic under  the  radical  sign.  (1).  Suppose  these  roots 
to  be  all  real,  and  denoted  by  a,  j3,  7,  8,  arranged  in  order  of 
increasing  magnitude,  and  we  have 

xy  = ±  a/ a  (x  -  a) (x  -  /3) (x  -  y)(x  -  §). 

Now  when  x  is  <  a,  y  is  real ;  when  x  >  a  and  <  /3,  y  is 
imaginary  ;  when  x  >  |3  and  <  7,  y  is  real ;  when  x>  7  and 
<  d,  y  is  imaginary ;  when  x>  S,  y  is  real. 


Asymptotes. 


325 


We  infer  that  the  curve  consists  of  three  branches,  extending 
to  infinity,  together 
with  an  oval  lying 
between  the  values 
(3  and  7  for  x. 

The  accompany- 
ing figure*  repre- 
sents such  a  curve. 

Again,  if  either 
the  two  greatest 
roots  or  the  two 
least  roots  become 
equal,  the  corres- 
ponding point  be- 
comes a  node. 

If  the  interme- 
diate roots  become  Fig-  39- 
equal,  the  oval  shrinks  into  a  conjugate  point  on  the  curve. 

If  three  roots  be  equal,  the  corresponding  point  is  a  cusp. 

If  two  of  the  roots  be  impossible  and  the  other  two  un- 
equal, the  curve  can  have  neither  an  oval  nor  a  double  point. 

If  the  sign  of  a  be  negative,  the  curve  has  but  one  real 
asymptote. 

261.  Asymptotes. — In  the  preceding  figure  the  student 
will  observe  that  to  each  asymptote  correspond  two  infinite 
branches ;  this  is  a  general  property  of  algebraic  curves,  of 
which  we  have  a  familiar  instance  in  the  common  hyperbola. 

By  the  student  who  is  acquainted  with  the  elementary 
principles  of  conical  projection  the  preceding  will  be  readily 
apprehended ;  for  if  we  suppose  any  line  drawn  cutting  a 
closed  oval  curve  in  two  points  at  which  tangents  are  drawn, 
and  if  the  figure  be  so  projected  that  the  intersecting  line  is 
sent  to  infinity,  then  the  tangents  will  be  projected  into 
asymptotes,  and  the  oval  becomes  a  curve  in  two  portions, 
each  having  two  infinite  branches,  a  pair  for  each  asymptote, 
as  in  the  hyperbola. 


*  The  figure  is  a  tracing  of  the  curve 

yxy1  +  ic%  =  (x  -  5)  (x  -  II)  (x  -  12). 


326 


On  Tracing  of  Curves. 


It  should  also  be  observed  that  the  points  of  contact  at 
infinity  on  the  asymptote  in  the  opposite  directions  along  it 
must  be  regarded  as  being  one  and  the  same  point,  since  they 
are  the  projection  of  the  same  point.  That  the  points  at 
infinity  in  the  two  opposite  directions  on  any  line  must  be 
regarded  as  a  single  point  is  also  evident  from  the  considera- 
tion that  a  right  line  is  the  limiting  state  of  a  circle  of  in- 
finite radius. 

The  property  admits  also  of  an  analytical  proof;  for  if 
the  asymptote  be  taken  as  the  axis  of  x,  the  equation  of  the 
curve  (Art.  204)  is  of  the  form 


y§\  +  (j>2  =  o, 


02 

Or  y  =  -  XL, 

0i 


where  02  is  at  least  one  degree  lower  than  <px  in  x  and  y. 

Now,  when  x  is  infinitely  great,  the  fraction  —  becomes  in 

0i 
general  infinitely  small,  whether  x  be  positive  or  negative ; 

and  consequently  the  axis  is  asymptotic  to  the  curve  in  both 
directions. 

262.  To  trace  the  Curve 

a3y2  =  bx*  +  x5, 
where  a  and  b  are  both  positive. 
Here       ya*  =  ±  x2  (x  +  b)%. 

**The  curve  is  symmetrical  with  respect 
to  the  axis  of  x,  and  has  two  infinite 
branches  ;  the  origin  is  a  double  cusp. 
The  shape  of  the  curve  is  exhibited  in  the 
figure  annexed. 

If  b  were  negative,  we  should  have 

ya%  =  ±  x2  (x  -  b)k 

Here  y  becomes  imaginary  for  values  of  x  less  than  b ; 
accordingly,  the  origin  is  a  conjugate  point  in  this  case  :  the 
curve  has  two  infinite  branches  as  in  the  former  case 

263.  To  trace  toe  Curve 

azy2  =  2abx2y  +  x5. 


Fig.  40. 


Form  of  Curve  near  a  Double  Point. 


327 


From  the  form  of  its  equation  we  see  that  the  origin  is 
a  point  of  os^-innexion  (Art.  251). 

Solving  for  y,  we  can  easily 
determine  any  number  of  points 
on  the  curve  we  please.  It  has 
two  infinite  branches  at  opposite 
sides  of  the  axis  of  x,  and  a  loop 
at  the  negative  side  of  that  axis, 
as  exhibited  in  the  figure. 

264.  To  trace  tbe  Curve 


x4,  +  x2 y2  +  yi  =  x  (ax2 


bf). 


Fig.  41. 


(1).  Let  a  and  b  have  the 
same  sign,  then  the  origin  is 
a  'triple  point,  having  for  its 
tangents  the  lines 

x  =  o,  x */a  +  y  /S  =  o, 

and        x  *ya  -  y  */b  =  o. 

Moreover,  since  the  curve 
has  no  real  asymptote,  it  is 
a  finite  or  closed  curve  with 
three  loops  passing  through  the  Fio«  42« 

origin  ;  and  it  is  easily  seen  that  its  shape 
is  that  represented  in  the  accompanying 
figure. 

(2).  If  a  and  b  have  opposite  signs,  the 
lines  represented  by  ax2  -  by2  =  o  become 
imaginary.  The  curve  in  this  case  consists 
of  a  single  oval  as  in  the  figure. 

This  and  the  preceding  figure  were 
traced  for  the  case  where  b  =  3a:  if  the 

value  of  -  be  altered,  the  shape  of  the  curve  Fig.  43. 

will  alter  at  the  same  time.     If  a  be  greater  than  b,  the 
curve  (2)  will  lie  inside  the  tangent  at  the  point  X. 

265.  Form  of  Curve  uear  a  DouMe  Point. — When- 
ever the  curve  has  a  node  or  a  cusp,  by  transforming  the 
origin  to  that  point,  the  shape  of  the  curve  for  the  branches 


328  On  Tracing  of  Curves. 

passing  through  the  point  admits  of  being  investigated  by  the 
method  explained  in  Arts.  250,  251.  It  is  unnecessary  to 
enter  into  detail  on  this  subject  here,  as  it  has  been  already 
discussed  in  the  articles  referred  to. 

266.  In  connexion  with  the  tracing  and  the  discussion  of 
curves  there  is  an  elementary  general  principle  which  may 
be  introduced  here. 

If  the  equation  of  a  curve  be  of  the  form 

LU  -  MM'  =  o, 

where  L,  M,  L',  Mf  are  each  functions  of  the  co-ordinates  x 
and  y,  the  curve  evidently  passes  through  all  the  points 
of  intersection  of  the  curves  represented  by  the  equations 
L  =  o  and  M  =  o ;  similarly  it  passes  through  the  intersec- 
tions of  L  =  o  and  M '  =  o ;  and  also  those  of  M  =  o  and 
If  =  o ;  and  of  L'  =  o  and  Mf  =  o.  Moreover,  if  L  and  L' 
become  identical,  the  points  of  intersection  coincide  in 
pairs,  and  the  equation  of  the  curve  becomes  of  the  form 
L*  -  MMr  =  o ;  which  represents  a  curve  touching  the  curves 
M  =  o,  M'  =  o,  at  their  points  of  intersection  with  the  curve 
L  =  o. 

This  principle  admits  of  easy  extension ;  but  as  the  subject 
belongs  properly  to  the  method  of  trilinear  co-ordinates,  it  is 
not  considered  necessary  to  enter  more  fully  into  it  here. 

267.  On  Tracing  Curves  given  in  Polar  Co-ordi- 
nates.— -The  mode  of  procedure  in  this  case  does  not  differ 
essentially  from  that  for  Cartesian  co-ordinates.  We  have 
already,  in  Arts.  206  and  207,  considered  the  method  of 
finding  the  asymptotes  and  asymptotic  circles  in  such  cases. 
It  need  scarcely  be  observed  that  the  number  and  variety  of 
curves  whose  discussion  more  properly  comes  under  the 
method  of  polar  co-ordinates  are  indefinite.  We  propose  to 
confine  our  attention  to  a  few  varieties  of  the  class  of  curves 
represented  by  the  equation 

rm  _  am  cog  mQ 

268.  On  the  Cnrves  rm  =  am  cos  mQ. — In  this  case, 
since  the  equation  is  unaltered  when  9  is  changed  into  -  Q, 
the  curve  is  symmetrical  with  respect  to  the  prime  vector : 
again,  when  6  =  o,  we  have  r  =  a ;  and  as  0  increases  from  zero 


Curves  of  the  Form  rm  =  am  cos  mO. 


329 


7T  -rrr 

to  — ,  r  diminislies  from  a  to  zero.     When  m  is  a  positive  in- 

2771 

teger,  it  is  easily  seen  that  the  curve  consists  of  m  similar  loops. 
There  are  many  familiar  curves  included  under  this 
equation.  Thus,  when  m  =  1 ,  we  have  r  =  a  cos  9,  which 
represents  a  circle:  again,  if  m  =  -  1,  the  equation  gives 
r  cos  0  =  a,  which  represents  a  right  line.  Also,  if  m=  2,  we 
have  r2  =  a2  cos  2  0,  a  Lemniscate  (Art.  210).  If  m  =  -  2,  we 
get  r2  cos  20  =  a2,  an  equilateral  hyperbola. 

If  w  =  -  we  get  r*  =  ah  cos  -,  whence  r  =  -  (1  +  cos  0),  a 

1  ft 

cardioid  (Ex.  4,  p.  232)  ;  with  m  =  — ,  itisr^  cos  -  =  a\   a 

parabola  (Ex.  1,  p.  231)  ;  and  so  on.  As  already  observed, 
if  we  change  m  into  -  m  we  get  a  new  curve,  inverse  of 
the  original.     Also,  the  reciprocal  polar  is  obtained  by  sub- 

stitutinsr  -    instead  of  m. 

m  +  1 

The  tangent  and  normal  can  be  immediately  drawn  at 
anycpoint  on  a  curve  of  this  class  by  aid  of  the  results  arrived 
at  in  Art.  190.  The  radius  of  curvature  at  any  point  has 
been  determined  in  Ex.  5,  Art.  235.  The  method  of  finding 
the  equations  of  the  successive  pedals,  both  positive  and 
negative,  has  been  also  already  explained. 

A  few  examples  in  the  case  of  fractional  indices  are  here 
added. 

Example  1. 

i    »    e 

rt  =  at  cos  -. 
3 
Here  when  6  =  o,  we  have  r  =  a, 
and  the  curve  cuts  the  prime  vector 
at  a  distance  OA  equal  to  a  :  again, 

when  6=-,r=  3-^: 

2  8 

0  =  tt,  r  =  -,  or  0B  =  -. 


also  when 


Fig.  44. 


The  shape  of  the  curve  is  given  in  the  accompanying 
figure.  This  curve  is  the  inverse  of  the  caustic  considered  in 
Example  18,  p.  277. 


33° 


On  Tracing  of  Curves. 


Ex.  2.  Ex.  3.  Ex.  4. 

ri  =  ai  cos- 9.         r*  =  a*  cos  -  9.         r*  =  at  cos  -  9. 
4  5  3 

In  Ex.  2,  as  0  increases  from  zero  to  1200,  r  diminishes 
from  a  to  zero :  when  9  increases 
from  1200  to  2400,  r  increases  from 
zero  to  a :  when  9  increases  from 
2400  to  3600,  r  diminishes  from  a 
to  zero.  By  assigning  negative 
values  to  9,  the  remaining  part  of 
the  curve  is  seen  to  be  symmetrical 
with  that  traced  as  above.  The 
same  result  plainly  follows  by  con- 
tinuing the  values  for  9  from  3600 
up  to  7200.  The  form  of  the  curve 
is  exhibited  in  the  annexed  figure.  Fig.  45. 

m  A 

In  Ex.  3,  according  as  cos  -  9  is  positive  or  negative,  we 

o 
get  equal  and  opposite  real  values,  or  imaginary  values,  for  r. 

Hence  it  is  easily  seen  that  for  values  of  9  between  ±  -  ir  the 

radius  vector  traces   out  two  symmetrical  portions   of  the 

curve :    again,  between  — -  tt  and  —  tt   we    get  two   other 

0 


Fig.  46.  Fig.  47. 

symmetrical  portions.     The  shape  is  that  given  in  the  former 
of  the  two  accompanying  figures. 


The  Limagon. 


33* 


The  latter  figure  represents  the  curve  in  Ex.  4 ;  it  consists 
of  five  symmetrical  portions  ranged  round  the  origin. 

The  results  above  stated  admit  of  generalization,  and  it 
can  be  shown,  without  difficulty,  that  in  general  the  curve 

r*  =  ofl  cos  —  consists  of  p  similar  portions  arranged  about 

q 
the  origin;  and  that  the  entire  curve  is  included  within  a 
circle  of  radius  a  when  p  is  positive,   but  lies  altogether 
outside  it  when  p  is  negative. 

Many  curves  can  be  best  traced  by  aid  of  some  simple 
geometrical  property.  We  shall  terminate  the  Chapter  with 
one  or  two  examples  of  such  curves. 

269.  The  Iiimagon. — The  inverse  of  a  conic  section 
with  respect  to  a  focus  is  called  a  Limacon.  From  the  polar 
equation  of  a  conic,  its  focus  being  origin,  it  is  evident  that 
the  equation  of  its  inverse  may  be  written  in  the  form 

r  =  a  cos  0  +  b, 

where  a  and  b  are  constants. 

It  is  easily  seen  that  -=-  is  the  eccentricity  of  the  conic. 

0 

The  curve  can  be  readily  traced  by  drawing  from  a  fixed 
point  on  a  circle  any  number  of  chords,  and  taking  off  a 
constant  length  on  each  of  these  lines,  measured  from  the 
circumference  of  the  circle. 

If  a  be  less  than  b,  the  curve  is  the  inverse  of  an  ellipse, 
and  lies  altogether  outside  the  circle. 

If  a  be  greater  than  b,  the 
curve  is  the  inverse  of  a  hy- 
perbola, and  its  form  can  be 
easily  seen  to  be  that  exhibited 
in  the  annexed  figure,  where 
OD  -  a  -  b9  and  the  point  0  is  a 
node  on  the  curve. 

If  b  =  a,  the  curve  becomes 
the  inverse  of  the  parabola, 
and  is  called  a  cardioid.  The 
inner  loop  disappears  in  this 
case,  and  the  origin  is  a  cusp 
on  the  curve.  Fig.  48. 


332 


On  Tracing  of  Curves. 


When  a  =  2b ,  the  Limacon  is  called  the  Trisectrix;  a 
curve  by  aid  of  which  any  given  angle  can  be  readily 
trisected. 

270.  The  Conchoid  of  UTicomedes. — If  through  any 
fixed  point  A  a  secant  PXAP  be 
drawn  meeting  a  fixed  right  line  LM 
in  B,  and  BP  and  BPX  be  taken 
each  of  the  same  constant  length; 
then  the  locus  of  P  and  Px  is  called 
the  conchoid. 

This  curve  is  easily  traced  from 
the  foregoing  geometrical  property, 
and  it  consists  of  two  branches, 
having  the  right  line  LM  for  a 
common  asymptote.  Moreover,  if 
the  perpendicular  distance  AB  of 
A  from  the  fixed  line  be  less  than 
BP,  the  curve  has  a  loop  with  a 
node  at  A,  as  in  the  annexed  figure. 

It  is  easily  seen  that  when 
AB  =  BP,  the  point  A  is  a  cusp 
on  the  curve  ;  and  when  AB  is 
greater  than  BP,  A  is  a  conjugate 
point. 

The  form  of  the  curve  in  the 
latter  case  is  represented  by  the  dotted  lines  in  the  figure. 

If  AB  =  a,  BP  =  b,  the  polar  equation  of  the  curve  is 
(r  ±  b)  cos  6  =  a. 

When  transformed  to  rectangular  co-ordinates,  this 
equation  becomes 

(xz  +  y*)(a-xy=  Vx\ 

The  method  of  drawing  the  normal,  and  finding  the 
centre  of  curvature,  at  any  point,  will  be  exhibited  in  the 
next  Chapter. 


Examples.  333 


Examples. 

1.  Trace  the  curve  y  —  (x  -  1)  (x  -  2)  (x  —  3),  and  find  the  position  of  its 
point  of  inflexion. 

2.  Trace  the  curve  y3  -  ^axy  +x3  =  o,  drawing  its  asymptote. 
This  curve  is  called  the  Folium  of  Descartes. 

3.  Trace  the  curve  a2x  =  y  (b2  +  x2),  and  find  its  points  of  inflexion,  and 
points  of  greatest  and  least  distance  from  the  axis  of  x. 

4.  If  an  asymptote  to  a  curve  meets  it  in  a  real  finite  point,  show  that  the 
corresponding  branch  of  the  curve  must  have  a  point  of  inflexion  on  it. 

5.  Find  the  position  of  the  asymptotes  and  the  form  of  the  curve 

a4  —  y4  +  iaxy2  =  o. 

6.  Show  that  the  curve  r  =  a  cos  20  consists  of  four  loops,  while  the  curve 
r  =  a  cos  30  consists  of  but  three.  Prove  generally  that  the  curve  r  =  a  cos  n0 
has  n  or  2n  loops  according  as  n  is  an  odd  or  even  integer. 

7.  Trace  the  curve 

y2  (x  -  a) (x  -  b)  =  c2 (x  +  a)  (x  +  b). 

8.  Show  that  the  curve  x*y2  +  x*  =  a2(x2  —  y2)  consists  of  two  loops  passing 
through  the  origin,  and  find  the  form  of  the  curve. 

9.  Trace  the  curve  y(x  +  a)4  =  b2x{x  +  c)2,  showing  the  positions  of  its 
asymptotes  and  infinite  branches. 

10.  Trace  the  curve  whose  polar  equation  is 

r  =  a  cos  0  +  b  cos  20, 

and  show  that  it  consists  of  four  loops  passing  through  the  origin. 

11.  Given  the  base  and  the  rectangle  under  the  sides  of  a  triangle,  find  the 
equation  of  the  locus  of  the  vertex  (an  oval  of  Cassini).  Exhibit  the  different 
forms  of  the  curve  obtained  by  varying  the  constants,  and  find  in  what  case  the 
curve  becomes  a  Lemniscate. 

12.  Trace  the  curve  y2  =  ax3  +  ^bx2  +  zcx  +  d,  and  find  its  points  of  greatest 
and  least  distance  from  the  axis  of  x. 

Show  that  two  of  these  points  become  imaginary  when  the  roots  of  the  cubic 
in  x  are  all  real. 

13.  Given  the  base  and  area  of  a  triangle,  prove  that  the  equation  of  the 
locus  of  the  centre  of  a  circle  touching  its  three  sides  is  of  the  form 

x2y  -  a  (x2  +  y2)  -  b2(y  -  a)  =  o. 


334  Examples. 

14.  Prove  that  all  curves  of  tlie  third  degree  are  reducible  to  one  or  other  of 
the  forms 

(1).    xy2  +  ey  =  axz  +  bx*  +  ex  +  d. 
(2).     xy  =  axz  +  bxz  +  ex  +  d. 
(3).     y2  =  axz  +  fo2  +  ex  +  d. 
(4) .    y    =  ax*  +  bx2  +  ex  +  d. 
Newton,  Enum.  Linear.  Ter.  Ordinis. 

15.  Prove  that  all  curves  of  the  third  degree  can  he  obtained  by  projection 
from  the  parabolas  contained  in  class  (3)  in  the  preceding  division.     [Newton.] 

For  every  cubic  has  at  least  one  real  point  of  inflexion :  accordingly,  if  the 
curve  be  projected  so  that  the  tangent  at  the  point  of  inflexion  is  projected  to 
infinity,  the  harmonic  polar  (Art.  223)  will  bisect  the  system  of  parallel  chords 
passing  through  this  point  at  infinity.  Hence  the  projected  curve  is  of  the 
class  (3).     [This  proof  is  taken  from  Chasles,  Histoire  de  la  Geometrie,  note  xx.] 

a92 

16.  Trace  the  curve  r  =  — ,  and  show  that  it  has  a  point  of  inflexion 

6l  —  1 

when  02  =  3 ;  find  also  its  asymptotes  and  asymptotic  circle. 

x 

17.  Trace  the  curve  y  =  a  sin-,  and  show  how  to  draw  its  tangent  at  any 

a 

point.     (This  is  called  the  curve  of  sines.) 

18.  The  base  of  a  triangle  is  fixed  in  position ;  find  the  equation  of  the  locus 
of  its  vertex,  when  the  vertical  angle  is  double  one  of  the  base  angles. 

Trace  the  locus  in  question,  finding  the  position  of  its  asymptote. 

19.  Show  geometrically  that  the  first  pedal  of  a  circle  with  respect  to  a 
point  on  its  circumference  is  a  cardioid. 

20.  Show  in  like  manner  that  the  Limacon  is  the  first  pedal  of  a  circle  with 
respect  to  any  point. 

2i.  Trace  the  curve 

f/*  +  2axy2  =  axz  +  xi, 

and  find  the  equations  of  its  asymptotes,  and  of  the  tangents  at  the  origin. 

Ind.  Civ.  Ser.  JSx.,  1876. 


(     335     ) 


CHAPTEE  XIX. 

ROULETTES. 

271.  Roulettes. — When  one  curve  rolls  without  sliding 
upon  another,  any  point  invariably  connected  with  the  rolling 
curve  describes  another  curve,  called  a  roulette. 

The  curve  which  rolls  is  called  the  generating  curve,  the 
fixed  curve  on  which  it  rolls  is  called  the  directing  curve,  or 
the  base,  and  the  point  which  describes  the  roulette,  the  tracing 
point.  We  shall  commence  with  the  simplest  example  of  a 
roulette :  viz.,  the  cycloid. 

272.  The  Cycloid. — This  curve  is  the  path  described  by 
a  point  on  the  circumference  of  a  circle,  which  is  supposed  to 
roll  upon  a  fixed  right  line. 

The  cycloid  is  the  most  important  of  transcendental 
curves,  as  well  from  the  elegance  of  its  properties  as  from  its 
numerous  applications  in  Mechanics. 

We  shall  proceed  to  investigate  some  of  the  most 
elementary  properties  of  the  curve. 

Let  LPO  be  any  position  of  the  rolling  circle,  P  the 
generating  point,  0  the  point  of 
contact  of  the  circle  with  the  fixed 
line.  Take  the  length  AO  equal 
to  the  arc  P0,  then,  from  the 
mode  of  generation  of  the  curve, 
A  is  the  position  of  the  generating 
point  when  in  contact  with  the  F*S-  5°» 

fixed  line ;  also,  if  AA!  be  equal  to  the  circumference  of  the 
circle,  A!  will  be  the  position  of  the  point  at  the  end  of  one 
complete  revolution  of  the  circle.  Bisect  AA!  in  D,  and 
draw  DB  perpendicular  to  it  and  equal  to  the  diameter  of 
the  circle,  then  B  is  evidently  the  highest  point  in  the 
cycloid.  Draw  PN  perpendicular  to  AA',  and  let  PN  =  y, 
AN  =x,  L  PCO  =  0,  OC  =  a,  and  we  get 

x  =  AO  -  NO  =  a{9-  sin  0),  y  =  PN =  a{i  -cos0).    (1) 


336  Roulettes, 

The  position  of  any  point  on  the  cycloid  is  determined  by 
these  equations  when  the  angle  0  is  known,  i.  e.  the  angle 
through  which  the  circle  has  rolled,  starting  from  the  position 
for  which  the  generating  point  is  upon  the  directing  line. 

2 73»  Cycloid  referred  to  its  Vertex. — It  is  often 
convenient  to  refer  the  cycloid  to  its  vertex  as  origin,  and  to 
the  tangent  and  normal  at  that  point  as  axes  of  co-ordinates. 
In  the  preceding  figure  let 

x  =  BN',    y  =  PN\    z.PCL  =  0'  =  Tr-0; 
then  we  have 

x  =  BN'  =  a  (6T  +  sin  00,    y  =  PN'  =  a  (i  -  cos  0').    (2) 

274.  Tangent  and  Normal  to  Cycloid. — It  can  be 

easily  seen  that  the  line  PO  is  normal  at  P  to  the  cycloid ; 
for  the  motion  of  each  point  on  the  circle  at  the  instant  is  one 
of  rotation  about  the  point  0,  i.  e.  each  point  may  be  regarded 
as  describing  at  the  instant  an  infinitely  small  circular*  arc 
whose  centre  is  at  0  :  and  hence  PO  is  normal  to  the  curve. 
This  result  can  also  be  established  from  the  values  of  x 
and  y  in  (1) :  for 

-|  =  0(1  -  cos  0),      J  =  asin0:  (3) 

.  dy        sin  0  ,0  D  T  ~ 

.  .  -f  = 7;  =  cot  -  =  cot  PLO ; 

dx      1  -  cos  0  2 

and,  accordingly,  PL  is  the  tangent,  and  PO  the  normal  to 
the  curve  at  P. 

dx         du 
Again,  if  we  square  and  add  the  values  of  -^  and  -^,  we 

obtain 

^Y=  a?  { (1  -  cos  Oy  +  sin2  0}  =  4a2  sin2  i  0; 


»  This  method  of  finding  the  normal  to  a  cycloid  is  due  to  Descartes,  and 
evidently  applies  equally  to  all  roulettes. 


The  Cycloid. 


337 


hence 


ds  .    0      -pn 

dQ  2 


(4) 


275.  Radius  of  Curvature  and  Evolute  of  Cycloid. 

— Let  p  denote  the  radius  of  curvature  at  the  point  P,  and 

lPOA  =  0  =  -; 

2 

,,  ds        ds  .    0  ^ 

tlien  '>  =  ^  =  2^  =  48sm5  =  2POs  (5) 

or  the  radius  of  curvature  is  double  the  normal.  From  this 
value  of  p  the  evolute  of  the  curve 
can  be  easily  determined.  For, 
produce  PO  until  OP'  =  OP,  then 
Pr  is  the  centre  of  curvature  be- 
longing to  the  point  P.  Again, 
produce  LO  until  00'  =  OL,  and 
describe  a  circle  through  0,  P/  and 
0' ;  this  circle  evidently  touches 
A  A',  and  is  equal  to  the  generating 
circle  LPO.  Fig.  51. 

Also,  the  arc  OP'  =  arc  OP  =  AO; 

.-.  arc  O'P'  =  O'P'O  -  P'O  =  AD  -  AO  =  OB  =  P'0\ 

Hence  the  locus  of  P'  is  the  cycloid  got  by  the  rolling  of 
this  new  circle  along  the  line 
B'G\  and  accordingly  the  evo- 
lute of  a  cycloid  is  another 
cycloid.  It  is  evident  that  the 
evolute  of  the  cycloid  ABA! 
is  made  up  of  the  two  semi- 
cycloids,  AB'  and  B'A,  as  in 
figure  51.  Conversely,  the 
cycloid  ABA'  is  an  involute  of 
the  cycloid  AB'A'. 

The  position  of  the  centre  of 
curvature  for  a  point  P  on  a 
cycloid  can  also  be  readily  de- 
termined geometrically,  as  fol- 
lows : — 

Suppose  Oi  a  point  on  the 
circle  infinitely  near  to   0,  and  take  00*  =  00*, 

z 


LstP' 


3}  8  Roulettes. 

be  the  centre  of  curvature  required,  and  draw  POl  and  P'02. 
Now  suppose  the  circle  to  roll  until  Ox  and  02  coincide,  then 
C02  becomes  perpendicular  to  AD,  and  POx  and  P'02  will 
lie  in  directum  (since  P'  is  the  point  of  intersection  of  two 
consecutive  normals  to  the  cycloid) .     Hence 

z  OCO,  =  L  POxQ  =  l  OPO,  +  l  oro1, 

since  each  side  of  the  equation  represents  the  angle  through 
which  the  circle  has  turned. 

But  L  OCOl  =  2  l  OP  Ol        (Euclid,  III.  20.) 

Hence  lOPO,  =  l  OP,01; 

.-.  P01  =  P'Ol] 

and  consequently  in  the  limit  we  have 

PO  =  P'O, 

as  before. 

"We  shall  subsequently  see  that  a  similar  method  enables 
us  to  determine  the  centre  of  curvature  for  a  point  in  any 
roulette. 

276.  length  of  Arc  of  Cycloid. — Since  AP'B'  (Fig.  51) 
is  the  evolute  of  the  cycloid  APB,  it  follows,  from  Art.  2  3  7,  that 
the  arc  AP"  of  the  cycloid  is  equal  in  length  to  the  line  PP\ 
or  to  twice  P'O ;  hence,  as  A  is  the  highest  point  in  the 
cycloid  AP'B',  it  follows  that  the  arc  AP'  measured  from  the 
highest  point  of  a  cycloid  is  double  the  intercept  P'O,  made 
on  the  tangent  at  the  point  by  the  tangent  at  the  highest 
point  of  the  curve. 

Hence,  denoting  the  length  of  the  arc  AP'  by  s,  we  have 

s  =  40  sin  P'OD  =  4#  sin  0.  (6) 

This  gives  the  intrinsic  equation  of  the  cycloid  (see  Art. 
24.2(a)).  Hence,  also,  the  whole  arc  AB'  is  four  times  the 
radius  of  the  generating  circle :  and  accordingly  the  entire 
length  ABA'  of  a  cycloid  is  eight  times  the  radius  of  its 
generating  circle. 

Again,  if  the  distance  of  P'  from  AA!  be  represented  by 
y,  we  shall  have 

P'O2  =  00'  xy  =  lay. 

Hence  s2  =  $FO~  =  Say.  fj) 


Epicycloids  and  Hypocycloids. 


33Q 


Fig.  S3- 

Their  forms  are  exhibited  in 


This  relation  is  of  importance  in  the  applications  of  the 
cycloid  in  Mechanics. 

Again,  since  AO  =  arc  OP\  if  we  represent  AO  by  v,  we 
have* 

v  =  2a<j>.  (8) 

277.  Trochoids.  —  In  general,  if  a  circle  roll  on  a 
right  line,  any  point  in  the. 
plane  of  the  circle  carried  round* 
with  it  describes  a  curve.  Such 
curves  are  usually  styled  tro- 
choids. "When  the  tracing 
point  is  inside  the  circle,  the 
locus  is  called  a  prolate  tro- 
choid ;  when  outside,  an  oblate, 
the  accompanying  figure. 

Their  equations  are  easily  determined;  for,  let  x,  y  be 
the  co-ordinates  of  a  tracing  point  P,  referred  to  the  axes 
AD,  and  AI  (A  being  the  position  for  which  the  moving 
radius  CP  is  perpendicular  to  the  fixed  line). 

Then,  if   CO  =  a,  CP  =  d,  L  OCP  =  0,  we  have 

x  =  AN=  AO-  OJST=aO-d  sin  0, ) 

(9) 
y  =  Pi\T  =  a  -d  cos  0.  ) 

278.  Epicycloidsf  and  Hypocycloids. — The  investi- 

*  This  is  called,  by  Professor  Casey,  the  tangential  equation  of  the  cycloid, 
and  by  aid  of  it  he  has  arrived  at  some  remarkable  properties  of  the  curve  ("  On 
a  New  Form  of  Tangential  Equation,"  Philosophical  Transactions,  1877).  "In 
general,  if  a  variable  line,  in  any  of  its  positions,  make  an  intercept  v  on  the  axis 
of  x,  and  an  angle  <p  with  it;  then  the  equation  of  the  line  is 

x  +  y  cot  <p  —  v  =  o  ; 

and  v,  0,  the  quantities  which  determine  the  position  of  the  line  may  be  called 
its  co-ordinates.  From  this  it  follows  that  any  relation  between  v  and  <j>,  such 
as 

v  =/(*), 

will  be  the  tangential  equation  of  a  curve,  which  is  the  envelope  of  the  line." 
For  applications,  the  reader  is  referred  to  Professor  Casey's  Memoir.  See  also 
Dub.  Exam.  Papers,  Graves,  Lloyd  Exhibition,  1847. 

f  I  have  in  this  edition  adopted  the  correct  definition  of  these  curves  as 
given  by  Mr.  Proctor  in  his  Geometry  of  Cycloids.  I  have  thus  avoided  the 
anomaly  existing  in  the  ordinary  definition,  according  to  which  every  epicycloid 

Z  2 


340 


Roulettes. 


gation  of  the  properties  of  the  cycloid  naturally  gave  rise  to 
the  discussion  of  the  more  general  case  of  a  circle  rolling  on  a 
fixed  circle.  In  this  case  the  curve  generated  by  any  point 
on  the  circumference  of  the  rolling  circle  is  called  an  epicycloid, 
or  a  hypocycloid,  according  as  the  rolling  circle  touches  the  outside, 
or  the  inside  of  the  circumference  of  the  fixed  circle.  We  shall 
commence  with  the  former  case. 

Let  P  be  the  position  of  the  generating  point  at  any  in- 
stant, A  its  position  when 
on  the  fixed  circle  ;  then 
the  arc  OA  =  arc  OP. 

Again,  let  C  and  (7  be 
the  centres  of  the  circles, 
a  and  b  their  radii, 
£ACO  =  0,  lOC'P=&\ 
then,  since  arc  OA  =  arc 
OP,  we  have  aO  =  bO. 

Now,  suppose  C  taken 
as  the  origin  of  rectangu- 
lar co-ordinates,  and  CA 
as  the  axis  of  x;  draw  PN 
and   GfL   perpendicular,  Fis-  54. 

and  PM parallel,  to  CA,  and  we  have 


x=CN=CL-JSfL  =(a+b)  cos  0  -  6  cos  (0  +  0'), 
y  =  PJSr=C'L-  C'M=  (a  +  b)smO-b  sin  (0  +  0'); 


a 


or,  substituting  -  0  for  0', 


x  =  {a  +  b)  cos  0  -  b  cos  —7—  0, 

a  +  b        > 

y  =  (a  +  b)  sin  0  -  b  sin  — =—  0. 

b         J 


(10) 


is  a  hypocycloid,  but  only  some  hypoeycloids  are  epicycloids.  While  according 
to  the  correct  definition  no  epicycloid  is  a  hypocycloid,  though  each  can  he  gene- 
rated in  two  ways,  as  will  be  proved  in  Art.  280. 


Epicycloids  and  Hype-cycloids.  341 

"When  the  radius  of  the  rolling  circle  is  a  submultiple  of 
that  of  the  fixed  circle,  the  tracing  point,  after  the  circle 
has  rolled  once  round  the  circumference  of  the  fixed  circle, 
evidently  returns  to  the  same  position,  and  will  trace  the 
same  curve  in  the  next  revolution.  More  generally,  if  the 
radii  of  the  circles  have  a  commensurable  ratio,  the  tracing 
point,  after  a  certain  number  of  revolutions,  will  return  to  its 
original  position :  but  if  the  ratio  be  incommensurable,  the 
point  will  never  return  to  the  same  position,  but  will  describe 
an  infinite  series  of  distinct  arcs.  As,  however,  the  suc- 
cessive portions  of  the  curve  are  in  every  respect  equal  to 
each  other,  the  path  described  by  the  tracing  point,  from 
the  position  in  which  it  leaves  the  fixed  circle  until  it  returns 
to  it  again,  is  often  taken  instead  of  the  complete  epicycloid, 
and  the  middle  point  of  this  path  is  called  the  vertex  of  the 
curve. 

In  the  case  of  the  hypocycloid,  the  generating  circle  rolls 
on  the  interior  of  the  fixed  circle,  and  it  can  be  easily  seen 
that  the  expressions  for  x  and  y  are  derived  from  those  in  (10) 
by  changing  the  sign  of  b  ;  hence  we  have 

x  =  (a  -  b)  cos  0  +  b  cos  — j~-  0, 

a-b        >  (II) 

y  =  (a  -  b)  sin  0  -  b  sin  —7—  6. 

The  properties  of  these  curves  are  best  investigated  by 
aid  of  the  simultaneous  equations  contained  in  formulas  (10) 
and  (11). 

It  should  be  observed  that  the  point  A,  in  Fig.  54,  is  a 
cusp  on  the  epicycloid ;  and,  generally,  every  point  in  which 
the  tracing  point  P  meets  the  fixed  circle  is  a  cusp  on  the 
roulette.  From  this  it  follows  that  if  the  radius  of  the  rolling 
circle  be  the  nth  part  of  that  of  the  fixed,  the  corresponding  epi- 
or  hypo-cycloid  has  n  cusps :  such  curves  are,  accordingly, 
designated  by  the  number  of  their  cusps  :  such  as  the  three- 
cusped,  four-cusped,  &c.  epi-  or  hypo-cycloids. 

Again,  as  in  the  case  of  the  cycloid,  it  is  evident  from 
Descartes'  principle  that  the  instantaneous  path  of  the  point  P 
is  an  elementary  portion  of  a  circle  having  0  as  centre ;  ac- 


342  Roulettes. 

cordingly,  the  tangent  to  the  path  at  P  is  perpendicular  to 
the  line  PO,  and  that  line  is  the  normal  to  the  curve  at  P. 
These  results  can  also  be  deduced,  as  in  the  case  of  the 
cycloid,  by  differentiation  from  the  expressions  for  x  and  y. 
We  leave  this  as  an  exercise  for  the  student. 

To  find  an  expression  for  an  element  ds  of  the  curve  at 
the  point  P;  take  0',  0" ',  two  points  infinitely  near  to  0  on 
the  circles,  and  such  that  00'  =  00"\  and  suppose  the  gene- 
rating circle  to  roll  until  these  points  coincide  :*  then  the 
lines  CO  and  CO"  will  lie  in  directum,  and  the  circle  will 
have  turned  through  an  angle  equal  to  the  sum  of  the  angles 
OCOf  and  OC'Of,\  hence,  denoting  these  angles  by  dO  and  d&, 
respectively,  we  have 

ds  =  OP  (dO  +  d&)  =  opfi  +  |)  <*0;  (12) 

since  dO'  =  T  dO. 

o 

279.    Radius   of  Curvature    of  an   Epicycloid. — 

Suppose  u)  to  be  the  angle  OSN  between  the  normal  at  P  and 
the  fixed  line  CA,  then 

o»=  (?OS-C'CS  =  ----0',  .'.  da,  =  -d0\i+^r 

22  [       20 

Hence,  if  p  be  the  radius  of  curvature  corresponding  to 
the  point  P,  we  get 

,.     *     0P^>.  (I3) 

r         d(s)  a  +  2b 

Accordingly,  the  radius  of  curvature  in  an  epicycloid  is 
in  a  constant  ratio  to  the  chord  OP,  joining  the  generating 
point  to  the  point  of  contact  of  the  circles. 


*  It  may  be  observed  that  O'O"  is  infinitely  small  in  comparison  with  00' ; 
bence  tbe  space  through  which  the  point  0  moves  during  a  small  displacement 
is  infinitely  small  in  comparison  with  the  space  through  which  Pmoves.  It  is 
in  consequence  of  this  property  that  0  may  be  regarded  as  being  at  rest  for  the 
instant,  and  every  point  connected  with  the  rolling  circle  as  having  a  circular 
motion  around  it. 


Double  Generation  of  Epicycloids  and  Hypocycloids.      343 


Fig.  55* 


280.  Double  Generation  of  Epicycloids  and  Hypo- 
cycloids. — In  an  Epicycloid,  it  can  be  easily  shown  that 
the  curve  can  be  generated  in  a  second  manner.  For, 
suppose  the  rolling  circle  in- 
closes the  fixed  circle,  and  join 
P,  any  position  of  the  tracing 
point,  to  0,  the  correspond- 
ing point  of  contact  of  the  two 
circles;  draw  the  diameter  OED, 
and  join  O'E  and  PD ;  connect 
C,  the  centre  of  the  fixed  circle, 
to  O,  and  produce  CO'  to  meet 
DP  produced  in  D',  and  describe 
a  circle  round  the  triangle  OPD'; 
this  circle  plainly  touches  the 
fixed  circle  ;  also  the  segments 
standing  on  OP,  OP,  and  00  are  obviously  similar ;  hence, 
since  OP  =  00'  +  O'P,  we  have 

arc  OP  =  arc  00'  +  arc  OP. 

If  the  arc  00' A  be  taken  equal  to  the  arc  OP,  we  have 
arc  Of  A  =  arc  OP ;  accordingly,  the  point  P  describes  the  same 
curve,  whether  we  regard  it  as  on  the  circumference  of  the 
circle  OPD  rolling  on  the  circle  OOE,  or  on  the  circumference 
of  OPD'  rolling  on  the  same  circle ;  provided  the  circles  each 
start  from  the  position  in  which  the  generating  point  coincides 
with  the  point  A.  Moreover,  it  is  evident  that  the  radius  of 
the  latter  circle  is  the  difference 
between  the  radii  of  the  other  two. 

Next,  for  the  Hypocycloid, 
suppose  the  circle  OPD  to  roll 
inside  the  circumference  of  OOE, 
and  let  C  be  the  centre  of  the 
fixed  circle  ;  join  OP,  and  pro- 
duce it  to  meet  the  circum- 
ference of  the  fixed  circle  in  O ; 
draw  O'E  and  PD,  join  CO, 
intersecting  PD  in  D',  and  de- 
scribe a  circle  round  the  triangle 
PD'O.  It  is  evident,  as  be- 
fore, that  this  circle  touches  the  Fig.  56. 


344 


Roulettes. 


larger  circle,  and  that  its  radius  is  equal  to  the  difference  be- 
tween the  radii  of  the  two  given  circles.  Also,  for  the  same 
reason  as  in  the  former  case,  we  have 


ar 


c  00'  =  arc  OP  +  arc  O'P. 


If  the  arc  OA  be  taken  equal  to  OP,  we  get  are  O'P 
=  arc  O'A  ;  consequently,  the  point  P  will  describe  the  same 
hypocycloid  on  whichever  circle  we  suppose  it  to  be  situated, 
provided  the  circles  each  set  out  from  the  position  for  which 
P  coincides  with  A. 

The  particular  case,  when  the  radius  of  the  rolling  circle  is 
half  that  of  the  fixed  circle,  may  be  noticed.  In  this  case  the 
point  D  coincides  with  C,  and  P  becomes  the  middle  point  of 
00',  and  A  that  of  the  arc  00'.  From  this  it  follows  im- 
mediately that  the  hypocycloid  described  by  P  becomes  the 
diameter  CA  of  the  fixed  circle.  This  result  will  be  proved 
otherwise  in  Art.  285. 

The  important  results  of  this  Article  were  given  by  Euler 
[Acta.Petrop.,  1781).  By  aid  of  them  all  epicycloids  can  be 
generated  by  the  rolling  of  a  circle  outside  another  circle; 
and  all  hypocycloids  by  the  rolling  of  a  circle  whose  radius 
is  less  than  half  that  of  the  fixed  circle. 

281.  Evolute  of  an  Epicycloid. — The  evolute  of  an 
epicycloid  can  be  easily 
seen  to  be  a  similar  epi- 
cycloid. 

For,  let  P  be  the  trac- 
ing point  in  any  position, 
A  its  position  when  on  the 
fixed  circle ;  join  P  to  0, 
the  point  of  contact  of  the 
circles,  and  produce  PO 

..,  -r^,      ^^2a  +  2b 
until  PPf  =  OP T, 

a  +  20 

then  P  is  the  centre  of 
curvature  by  (13) ;  hence 

a 


or  =op 

a  +  20  Fig.  57. 

Next,  draw  P'O'  perpendicular  to  P'O;  circumscribe  the 


Evolute  of  Epicycloid.  345 

triangle  OP'O'  by  a  circle ;  and  describe  a  circle  with  C  as 
centre,  and  CO'  as  radius :  it  evidently  touches  the  circle  OP'O'. 

Then     00' :  OE  =  OP' :  OP  =  a:  a  +  2b  =  CO  :  CE; 

.\   CO-00':CE--OE  =  CO:CE, 

or  CO'  :CO=CO  :  G# ; 

that  is,  the  lines  CE,  CO,  and  CO'  are  in  geometrical  pro- 
portion. 

Again,  join  C  to  B',  the  vertex  of  the  epicycloid  ;  let  CB' 
meet  the  inner  circle  in  D,  and  we  have 

arc  0'B:slvgOB=  CO':  C0=  CO  :  CE  =  O'O  :EO 

=  arc  P'O':  arc  OQ. 

But  arc  OB  =  arc  OQ  :         .-.  arc  07)  =  arc  P'C 

Accordingly,  the  path  described  by  P'  is  that  generated  by  a 
point  on  the  circumference  of  the  circle  OP'O'  rolling  on  the 
inner  circle,  and  starting  when  P'  is  in  contact  at  D.  Hence 
the  evolute  of  the  original  epicycloid  is  another  epicycloid. 
The  form  of  the  evolute  is  exhibited  in  the  figure. 

Again,  since  CO  :  OE  =  CO' :  O'O,  the  ratio  of  the  radii 
of  the  fixed  and  generating  circles  is  the  same  for  both  epicy- 
cloids, and  consequently  the  evolute  is  a  similar  epicycloid. 

Also,  from  the  theory  of  evolutes  (Art.  237),  the  line 
PPr  is  equal  in  length  to  the  arc  P'A  of  the  interior  epicy- 
cloid ;  or  the  length  of  P'A,  the  arc  measured  from  the 
vertex  A  of  the  curve,  is  equal  to 

2J^3op'  =  2op'^  =  2or^ 

a        Ur      2Ur  CO         Ur  CO'' 

Hence,  the  length*  of  any  portion  of  the  curve  measured  from 
its  vertex  is  to  the  corresponding  chord  of  the  generating  circle  as 
twice  the  sum  of  the  radii  of  the  circles  to  the  radius  of  the  fixed 
circle. 

*  The  length  of  the  arc  of  an  epicycloid,  as  also  the  investigation  of  its 
evolute,  were  given  hy  Newton  (Principia,  Lib.  1.,  Props.  49,  50): 


346 


Roulettes. 


"With  reference  to  the  outer  epicycloid  in  Fig.  57,  this 
gives 


arc  P&  =  2PE  . 


ccr 

CO' 


(i4) 


The  corresponding  results  for  the  hypocycloid  can  be 
found  by  changing  the  sign  of  the  radius  b  of  the  rolling 
circle  in  the  preceding  formulse. 

The  investigation  of  the  properties  of  these  curves  is  of 
importance  in  connexion  with  the  proper  form  of  toothed 
wheels  in  machinery. 

282.  Pedal  of  Epicycloid. — The  equation  of  the  pedal, 
with  respect  to  the  centre  of  the 
fixed  circle,  admits  of  a  very 
simple  expression.  For  let  P  be 
the  generating  point,  and,  as  be- 
fore, take  arc  OA  =  arc  OP,  and 
make  AB  =  900.  Join  CA,  CB, 
CP,  and  draw  CN  perpendicular 
to  DP.  Let  lPBO  =  $,l  BCN 
=  <d,lACO  =  0,  CJST  =  p. 

Then  since  AO  =  PO,  we  have 


aO  =  2b(p ; 


»  =  ?%. 

a 


Again,  oj  =  go° -ACN=0  +  <ji 

(       2bs 
f(i  +- 


Fig.  58. 


hence 


Also 


tf>  = 


(th) 


a  +  2b 


CN=  CBsin<p; 
.*.  p  =  (a  +  2b)  sin 


aw 


a  +  2b' 


(15) 


(16) 


which  is  the  equation  of  the  required  pedal. 

2  83 .  Equation  of  Epicycloid  in  terms  of  r  and  p. — 

Again,  draw  OL  parallel  to  BN,  and  let  CP  =  r,  and  we  have 


r2  -  p2 


PN2  =  OL2  =  OC2  -  CL2  =  a2- 


a  +  2b 


P 


2 . 


Epitrochoids  and  Hypotrochoids.  347 

hence  r2  =  a2  +  -7-^ — 77/ P2-  (17) 

(#  +  zof  v    ' 

Also,  from  (16)  it  is  plain  that  the  equation  of  DN,  the  tan- 
gent to  the  epicycloid  (referred  to  CB  and  CA  as  axes  of  x 
and  y  respectively),  is 

a?  cos  10  +  y  sina>  =  (a  +  2b)  sin 7.  (18) 

u  v  J        a  +  2b  v     7 

The  corresponding  formulae  for  the  hypocycloid  are 
obtained  by  changing  the  sign  of  b  in  the  preceding  equa- 
tions. 

Again,  it  is  plain  that  the  envelope  of  the  right  line  re- 
presented by  equation  (18)  is  an  epicycloid.  And,  in  general, 
the  envelope  of  the  right  line 

x  cos  ay  +  y  sin  id  =  Jc  sin  mw, 

regarding  w  as  an  arbitrary  parameter,  is  an  epicycloid,  or  a 
hypocycloid,  according  as  m  is  less  or  greater  than  unity.  For 
examples  of  this  method  of  determining  the  equations  of  epi- 
and  hypo-cycloids  the  student  is  referred  to  Salmon's  Higher 
Plane  Curves,  Art.  310. 

284.  JEpitrochoids  and  Hypotrochoids. — In  general, 
when  one  circle  rolls  on  another,  every  point  connected  with 
the  rolling  circle  describes  a  distinct  curve.  These  curves  are 
called  epitrochoids  or  hypotrochoids,  according  as  the  rolling 
circle  touches  the  exterior  or  the  interior  of  the  fixed  circle. 

If  d  be  the  constant  distance  of  the  generating  point  from 
the  centre  of  the  rolling  circle,  there  is  no  difficulty  in 
proving,  as  in  Art.  278,  that  we  have  in  the  epitrochoid  the 
equations 

x  =  (a  +  b)  cos  0  -  d  cos  — - —  0, 

a  +  b    >         (19) 

y  =  (a  +  b)  sin  9  -  d  sin  — - —  9. 


348 


Roulettes. 


In  the  case  of  the  hypotrochoid,  changing  the  signs  of  b 
and  d,  we  obtain 


x 


_       _         a  -  b  n  ~\ 
=  (a-  b)  cos  6  +  d  cos  —r~  u, 


.    a  —  b  „ 
y  =  (a  -  b)  sm  6  -  d  sm  — - —  U. 


\ 


(20) 


J 


In  the  particular  case  in  which  a  =  2b,  i.e.  when  a  circle 
rolls  inside  another  of  double  its  diameter,  equations  (20) 
become 

x  =  (b  +  d)  cos  6,     y  =  (b  -  d)  sin  0 ; 
and  accordingly  the  equation  of  the  roulette  is 

f 


X' 


+  - 


{b  +  dy      (b-d) 


=  1 ; 


which  represents  an  ellipse  whose  semi-axes  are  the  sum  and 
the  difference  of  b  and  d. 

This  result  can  also  be  established  geometrically  in  the 
following  manner : — 

285.  Circle  rolling  inside  another  of  double  its 
Diameter. — Join  Cx  and  0  to  any 
point  L  on  the  circumference  of  the 
rolling  circle,  and  let  CyL  meet  the 
fixed  circumference  in  A ;  then  since 
L  OCL  =  20CXA,  and  OCx  =  2OC,  we 
have  arc  OA  =  arc  OL ;  and,  accord- 
ingly, as  the  inner  circle  rolls  on  the 
outer  the  point  L  moves  along  CXA. 
In  like  manner  any  other  point  on 
the  circumference  of  the  rolling  circle 
describes,  during  the  motion,  a  dia- 
meter of  the  fixed  circle. 

Again,  any  point  P,  invariably  connected  with  the  rolling 
circle,  describes  an  ellipse.  For,  if  L  and  M  be  the  points  in 
which  CP  cuts  the  rolling  circle,  by  what  has  been  just 
shown,  these  points  move  along  two  fixed  right  lines  CXA 
and  &B,  at  right  angles  to  each  other.     Accordingly,  by  a 


Epitrochoids  and  Hypotrochoids. 


349 


well-known  property  of  the  ellipse,  any  other  point  in  the 
line  LM  describes  an  ellipse. 

The  ease  in  which  the  outer  circle  rolls  on  the  inner  is 
also  worthy  of  separate  consideration. 

286.  Circle  rolling  on  another  inside  it  and  of 
naif  its  Diameter. — In  this  case,  any  diameter  of  the  rolling 
circle  always  passes  through  a  fixed  point,  which  lies  on  the 
circumference  of  the  inner  circle. 

For,  let  CiL  and  CJL  be  any  two  positions  of  the  moving 
diameter,  Cx  and  02  being  the  corresponding  positions  of  the 
centre  of  the  rolling  circle  :  0  and  02  the  corresponding  posi- 
tions of  the  point  of  contact  of  the  circles.  Now,  if  the  outer 
circle  roll  from  the  former  to  the  latter  position,  the  right 
lines  (7i#2  and  C02  will  coincide  in 
direction,  and  accordingly  the  outer 
circle  will  have  turned  through  the 
angle  C2O2C1;  consequently,  the  mov- 
ing diameter  will  have  turned 
through  the  same  angle ;  and  hence 
L  CJLCx  =  lCzO%Ci',  therefore  the 
point  L  lies  on  the  fixed  circle,  and 
the  diameter  always  passes  through 
the  same  point  on  this  circle. 

Again,   any  right    line    connected 
with  the  rolling  circle  will  alicays  touch 
a  fixed  circle. 

For,  let  BE  be  the  moving  line  in  any  position,  and  draw 
the  parallel  diameter  AB\  let  fall  CiF  and  LM  perpendicular 
to  BE.  Then,  by  the  preceding,  AB  always  passes  through 
a  fixed  point  L ;  also  LM  =  CXF=  constant ;  hence  BE  always 
touches  a  circle  having  its  centre  at  L. 

Again,  to  find  the  roulette  described  by  any  carried  point 
Pi.  The  right  line  PiCh  as  has  been  shown,  always  passes 
through  a  fixed  point  L ;  consequently,  since  CJ?X  is  a  con- 
stant length,  the  locus  of  Pi  is  a  Limagon  (Art.  269).  In  like 
manner,  any  other  point  invariably  connected  with  the  outer 
circle  describes  a  Limacon ;  unless  the  point  be  situated  on 
the  circumference  of  the  rolling  circle,  in  which  case  the 
locus  becomes  a  cardioid. 


Fig.  60. 


350  Examples — Roulettes. 

i.  "When  the  radii  of  the  fixed  and  the  rolling  circles  hecome  equal,  prove 
geometrically  that  the  epicycloid  becomes  a  cardioid,  and  the  epitrochoid  a 
Limacon  (Art.  269). 

2.  Prove  that  the  equation  of  the  reciprocal  polar  of  an  epicycloid,  with 
respect  to  the  fixed  circle,  is  of  the  form 

r  sin  ma  =  const. 

3.  Prove  that  the  radius  of  curvature  of  an  epicycloid  varies  as  the  perpen- 
dicular on  the  tangent  from  the  centre  of  the  fixed  circle. 

4.  If  a  =  45,  prove  that  the  equation  of  the  hypocycloid  becomes 

#§  +  y%  =  flf . 

5.  Find  the  equation,  in  terms  of  r  and^?,  of  the  three-cusped  hypocycloid ; 
i.  e.  "when  a  =  3b.  Ans.  r2  =  a2  —  Sp2. 

6.  Find  the  equation  of  the  pedal  in  the  same  curve. 

Ans.  p  =  b  sin  3a. 

7.  In  the  case  of  a  curve  rolling  on  another  which  is  equal  to  it  in  every 
respect,  corresponding  points  being  in  contact,  prove  that  the  determination  of 
the  roulette  of  any  point  P  is  immediately  reduced  to  finding  the  pedal  of  the 
rolling  curve  with  respect  to  the  point  P. 

8.  Hence,  if  the  curves  be  equal  parabolas,  show  that  the  path  of  the  focus 
is  a  right  line,  and  that  of  the  vertex  a  cissoid. 

9.  In  like  manner,  if  the  curves  be  equal  ellipses,  show  that  the  path  of  the 
focus  is  a  circle,  and  that  of  any  point  is  a  bicircular  quartic. 

10.  In  Art.  285,  prove  that  the  locus  of  the  foci  of  the  ellipses  described  by 
the  different  points  on  any  right  line  is  an  equilateral  hyperbola. 

ir.  A  is  a  fixed  point  on  the  circumference  of  a  circle ;  the  points  L  and  M 
are  taken  such  that  arc  AL  —  m  arc  AM,  where  m  is  a  constant ;  prove  that  the 
envelope  of  LM  is  an  epicycloid  or  a  hypocycloid,  according  as  the  arcs  AL  and 
AM  axe  measured  in  the  same  or  opposite  directions  from  the  point  A. 

12.  Prove  that  LM,  in  the  case  of  an  epicycloid,  is  divided  internally  in  the 
ratio  m  :  1,  at  its  point  of  contact  with  the  envelope  ;  and,  in  the  hypocycloid, 
externally  in  the  same  ratio. 

13.  Show  also  that  the  given  circle  is  circumscribed  to,  or  inscribed  in,  the 
envelope,  according  as  it  is  an  epicycloid  or  hypocycloid. 

14.  Prove,  from  equation  (14),  that  the  intrinsic  equation  of  an  epicycloid  is 

±b  (a  +  b)     .        ad> 
s  = sin 


a  +  2b 

where  s  is  measured  from  the  vertex  of  the  curve. 

15.  Hence  the  equation  s  —  I  sin  nd>  represents  an  epicycloid  or  a  hypo- 
cycloid, according  as  n  is  less  or  greater  than  unity. 


Centre  of  Curvature  of  an  Epitrochoid  or  Hypotrochoid.    35 


16.  In  an  epitrochoid,  if  the  distance,  d,  of  the  moving  point  from  the  centre 
of  the  rolling  circle  be  equal  to  the  distance  between  the  centres  of  the  circles, 
prove  that  the  polar  equation  of  the  locus  becomes 


r  =  2  («  +  b)  cos 


ad 


a  +  2b 


17.  Hence  show  that  the  curve 

r  =  a  sin  mO 

is  an  epitrochoid  when  m  <  1,  and  a  hypotrochoid  when  m  >  1. 

This  class  of  curves  was  elaborately  treated  of  by  the  Abbe  Grandi  in  the 
Philosophical  Transactions  for  1723.  He  gave  them  the  name  of  "  Rhodonese," 
from  a  fancied  resemblance  to  the  petals  of  roses.  See  also  Gregory's  Examples 
on  the  Differential  and  Integral  Calculus,  p.  183. 

For  illustrations  of  the  beauty  and  variety  of  form  of  these  curves,  as  well  as 
of  epitrochoids  and  hypotrochoids  in  general,  the  student  is  referred  to  the  admi- 
rable figures  in  Mr.  Proctor's  Geometry  of  Cycloids. 


287.  Centre 
Mypotrochoid.- 


of  Curvature  of  an  Epitrochoid  or 

-The  position  of  the  centre  of  curvature  for 


any  point  of  an  epitrochoid  can  be  easily- 
found  from  geometrical  considerations.  For, 
let  Ci  and  C2  be  the  centres  of  the  rolling 
and  the  fixed  circles,  P2  the  centre  of  cur- 
vature of  the  roulette  described  by  Pi ;  and, 
as  before,  let  Ox  and  02  be  two  points  on  the 
circles,  infinitely  near  to  0,  such  that  00x 
=  002.  Now,  suppose  the  circle  to  roll  until 
Ox  and  02  coincide;  then  the  lines  CX0X 
and  C202  will  lie  in  directum,  as  also  the 
lines  P1O1  and  P202  (since  P2  is  the  point  Fig.  61. 

of  intersection  of  two  consecutive  normals  to 
the  roulette). 

Hence       L  0CX0X  +  L  OCA  =  L  0PX0X  +  L  OP202, 

since  each  of  these  sums  represents  the  angle  through  which 
the  circle  has  turned. 


Again,  let    z  CxOPx  =  0,     00x  =  002  =  ds ; 
then 


*Oftft-*p    <«MV-^. 


(JJr  i  C/P2 


35 2  Roulettes. 

consequently  we  have 

m+m'm*[m+Gp}        (2I) 

Or,  if  OPx  =  r1}     OP2  =  r2, 

ii  (\       i 

-  +  T  =  COS  d>    —  +  — 

«      6  r\n      r2 

From  this,  equation  r2,  and  consequently  the  radius  of  curva- 
ture of  the  roulette,  can  be  obtained  for  any  position  of  the 
generating  point  Pi. 

If  we  suppose  Pi  to  be  on  the  circumference  of  the  rolling 

OP 

circle,  we  get  cos  0  =         *  ;  whence  it  follows  that 

2  (J  0 1 

OP,  =  — °—-  OPl9 

a  +  2b 

which  agrees  with  the  result  arrived  at  in  Art.  279. 

288.  Centre  of  Curvature  of  any  Roulette. — The 

preceding  formula  can  be  readily  extended  to  any  roulette  :  for 
if  (7i  and  C2  be  respectively  the  centres  of  curvature  of  the 
rolling  and  fixed  curves,  corresponding  to  the  point  of  contact  0, 
we  may  regard  OOi  and  002  as  elementary  arcs  of  the  circles 
of  curvature,  and  the  preceding  demonstration  will  still 
hold. 

Hence,  denoting  the  radii  of  curvature  0&  and  0C2  by 
pi  and  p2,  we  shall  have 

—  +  —  =  cos  &  I  —  +  -  ].  (22) 

Pi     p%  r\ri     r2J 

It  can  be  easily  seen,  without  drawing  a  separate  figure, 
that  we  must  change  the  sign  of  p2  in  this  formula  when  the 
centres  of  curvature  lie  at  the  same  side  of  0. 

It  may  be  noted  that  P1  is  the  centre  of  curvature  of  the 
roulette  described  by  the  point  P2,  if  the  lower  curve  be  sup- 
posed to  roll  on  the  upper  regarded  as  fixed. 

289.  Greometrical  Construction*  for  the  Centre  of 

*  This  beautiful  construction,  and  also  the  formula  (22)  on  which  it  is  based, 
were  given  by  M.  Savary,  in  his  Lemons  des  Machines  a  V  Ecole  Poly  technique. 
See  also  Leroy's  Geome'trie  Descriptive,  Quatrieme  Edition,  p.  347. 


Construction  for  Centre  of  Curvature. 


353 


Curvature  of  a  Roulette. — The  formula  (22)  leads  to 
a  simple  and  elegant  construction  for  the  centre  of  curva- 
ture P2. 

We  commence  with  the  case  when  the  base  is  a  right 
line,    as  represented  in  the  accom- 
panying figure. 

Join  P,  to  Ci,  the  centre  of  curva- 
ture of  the  rolling  curve,  and  draw 
OiV"  perpendicular  to  OP„  meeting 
P,C,  in  JSf;  through  iV  draw  NM 
parallel  to  OCly  and  the  point  P2  in 
which  it  meets  OP,  is  the  centre  of 
curvature  required. 

For,  equation  (22)  becomes  in 
this  case 


A 


oc1  =  cos1'\oF>  +  op> 


whence  we  get 
PiP2      __ 
OP,.  OP, 


JSTP, 


JVC,  .  OP, ' 


OC,  sin  C,  ON     NC,  sin  C,NO 
P1P2=NP1 

"  op,    jyc,9 

and,  accordingly,  the  line  NP,  is  parallel  to  OC,.        Q.  E.  D. 

The  construction  in  the  general  case  is  as  follows  : — 

Determine  the  point  N  as  in  the  former 
case,  and  join  it  to  C2,  the  centre  of  curva- 
ture of  the  fixed  curve,  then  the  point  of 
intersection  of  NC,  and  P,0  is  the  required 
centre  of  curvature. 

This  is  readily  established  ;  for,  from 
the  equation 

+  JL_00B^_L_  +  _L. 

C,C,  COS  (f>P,P, 


oc, 

we  get 


oc.oc, 
(he*  OP^ 

OC,  'P,P, 


OP,.  OP, 

OC,  cos  $ 

op,  • 

2  A 


Fig.  63. 


354 

But,  as  before, 

00,  cos  <p  = 
hence 


Roulettes. 


0,N.OP,  OC,  cost     WO, 


NP, 


OP, 


NP,' 


Q,Q2    0I\      NC^ 

OC2'P,P2~  NP, 


Consequently,  by  the  well-known  property  of  a  transversal 
cutting  the  sides  of  a  triangle,  the  points  02,  Pz,  and  N  are 
in  directum. 

The  modification  in   the  construction  when  the  rolling 
curve  is  a  right  line  can  be  readily  supplied  by  the  student. 

290.  Circle  of  Inflexions. — The  following  geometrical 
construction  is  in  many  cases  more  q 

useful  than  the  preceding. 

On  the  line  00,  take  OB,  such 
that 


+ 


OB,      00,      OCa' 

and  on  OB,  as  diameter  describe  a 
circle.  Let  E,  be  its  point  of  inter- 
section with  OPi,  then  we  have 


cos  <p 


OE, 
OB? 


and  formula  (22)  becomes 


1  1 

+ 


(23) 


OP,      OPo      OB,  cos  <j>      OE,' 
Hence,  if  the  tracing  point  P,  lie  on  the  circle*  OE,B„ 


*  This  theorem  is  due  to  La  Hire,  who  showed  that  the  element  of  the 
roulette  traced  hy  any  point  is  convex  or  concave  with  respect  to  the  point  of 
contact,  0,  according  as  the  tracing  point  is  inside  or  outside  this  circle.     (See 


Envelope  of  a  Carried  Curve.  355 

the  corresponding  value  of  OP2  is  infinite,  and  consequently 
Pi  is  a  point  of  inflexion  on  the  roulette. 

In  consequence  of  this  property,  the  circle  in  question  is 
called  the  circle  of  inflexions,  as  each  point  on  it  is  a  point  of 
inflexion  on  the  roulette  which  it  describes. 

Again,  it  can  be  shown  that  the  lines  PiP2,  P\0  and  Px  Ex 
are  in  continued  proportion ;  as  also  CXC2,  Ci09  and  (7iA- 
For,  from  (23)  we  have 

P.P.  1 


? 


OPx .  OP2      OEx' 
Hence  PXP2  :P10=  OP,:  OE, 

.'.  P1P2  :P10  =  P,P2  -  OP2 :  P,0  -  OEx  =  PxO :  P1E1.     (24) 
In  the  same  manner  it  can  be  shown  that 

C1C2:ClO  =  &O:  Oft.  (25) 

In  the  particular  case  where  the  base  is  a  right  line,  the 
circle  of  inflexions  becomes  the  circle  described  on  the  radius 
of  curvature  of  the  rolling  curve  as  diameter. 

Again,  if  we  take  OD2  =  ODly  we  shall  have,  by  describing 
a  circle  on  OD2  as  diameter, 

02Oi  1  Cs2(J  =  (J2(-s  '.  \j2jJ2  5 

and  also  P2PX :  P20  =  P20  :  P2E2.  (26) 

The  importance  of  these  results  will  be  shown  further  on. 

291.  Envelope  of  a  Carried  Curve. — We  shall  next 
consider  the  envelope  of  a  curve  invariably  connected  with  the 
rolling  curve,  and  carried  with  it  in  its  motion. 

Since  the  moving  curve  touches  its  envelope  in  each  of  its 


Memoires  de  V 'Academie  des  Sciences,  1706.)  It  is  strange  that  this  remarkable 
result  remained  almost  unnoticed  until  recent  years,  when  it  was  found  to 
contain  a  key  to  the  theory  of  curvature  for  roulettes,  as  well  as  for  the 
envelopes  of  any  carried  curves.  How  little  it  is  even  as  yet  appreciated  in 
this  country  will  he  apparent  to  any  one  who  studies  the  most  recent  produc- 
tions on  roulettes,  even  by  distinguished  British  Mathematicians. 

2  A  2 


356  Roulettes. 

positions,  the  path  of  its  point  of  contact  at  any  instant  must 
be  tangential  to  the  envelope ;  hence  the  normal  at  their 
common  point  must  pass  through  0,  the  point  of  contact  of 
the  fixed  and  rolling  curves. 

In  the  particular  case  in  which  the  carried  curve  is  a 
right  line,  its  point  of   contact  with 
its  envelope  is  found  by  dropping   a 
perpendicular  on  it  from  the  point  of 
contact  0. 

For  example,  suppose  a  circle  to 
roll  on  any  curve :  to  find  the  envelope* 
of  any  diameter  PQ  : — 

From  0  draw  OJV  perpendicular 
to  PQ,  then  JV,  by  the  preceding,  is  Fi    65 

a  point  on  the  envelope. 

On  OC  describe  a  semicircle;  it  will  pass  through  N, 
and,  as  in  Art.  286,  the  arc  ON =  arc  OP  =  OA,  if  A  be 
the  point  in  which  P  was  originally  in  contact  with  the 
fixed  curve.  Consequently,  the  envelope  in  question  is  the 
roulette  traced  by  a  point  on  the  circumference  of  a  circle 
of  half  the  radius  of  the  rolling  circle,  having  the  fixed  curve 
AO  for  its  base. 

For  instance,  if  a  circle  roll  on  a  right  line,  the  envelope  of 
any  diameter  is  a  cycloid,  the  radius  of  whose  generating  circle 
is  half  that  of  the  rolling  circle. 

Again,  if  a  circle  roll  on  another,  the  envelope  of  any 
diameter  of  the  rolling  circle  is  an  epicycloid,  or  a  hypocycloid. 

Moreover,  it  is  obvious  that  if  two  carried  right  lines  be 
parallel,  their  envelopes  will  be  parallel  curves.  For  ex- 
ample, the  envelope  of  any  right  line,  carried  by  a  circle 
which  rolls  on  a  right  line,  is  a  parallel  to  a  cycloid,  i.e.  the 
involute  of  a  cycloid. 

These  results  admit  of  being  stated  in  a  somewhat  different 
form,  as  follows : 

If  one  point,  A,  in  a  plane  area  move  uniformly  along  a 
right  line,  while  the  area  turns  uniformly  in  its  own  plane, 
then  the  envelope  of  any  carried  right  line  is  an  involute  to  a 
cycloid.     If  the  carried  line  passes  through  the  moving  point 

*  The  theorems  of  this  Article  are,  I  believe,  due  to  Chasles  :  see  bis  Sistoire 
de  La  Geometrie,  p.  69. 


Centre  of  Curvature  of  the  Envelope  of  a  Carried  Curve.    357 

A,  its  envelope  is  a  cycloid.  Again,  if  the  point  A  move 
uniformly  on  the  circumference  of  a  fixed  circle,  while  the 
area  revolves  uniformly,  the  envelope  of  any  carried  right 
line  is  an  involute  to  either  an  epi-  or  hypo-cycloid.  If  the 
carried  right  line  passes  through  A,  its  envelope  is  either  an 
epi-  or  hypo-cycloid. 

292.  Centre  of  Curvature  of  the  Envelope  of  a 
Carried  Curve. — Let  aj>i  represent  a 
portion  of  the  carried  curve,  to  which  Om 
is  normal  at  the  point  m ;  then,  by  the 
preceding,  m  is  the  point  of  contact  of  a^bx 
with  its  envelope. 

Now,  suppose  a2b2  to  represent  a  por- 
tion of  the  envelope,  and  let  Px  be  the 
centre  of  curvature  of  a}blr  for  the  point  m, 
and  P2  the  corresponding  centre  of  cur- 
vature of  a2b2. 

As  before,  take  Ox  and  02  such  that 
00,  =  002,  and  join  PxOx  and  P202. 
Again,  suppose  the  curve  to  roll  until 
Oi  and  0%  coincide;  then  the  lines  P1O1 
and  P202  will  come  in  directum,  as  also 
the  lines  OyC,  and  OzC2 ;  and,  as  in  Art. 
288,  we  shall  have 


Fig.  66. 


z(71  +  z<72  =  zP1  +  lP2\ 


and  consequently 


oa 


+ 


m =  oos  *  \m +  m)- 


(27) 


From  this  equation  the  centre  of  curvature  of  the  enve- 
lope, for  any  position,  can  be  found.  Moreover,  it  is  obvious 
that  the  geometrical  constructions  of  Arts.  289,  290,  equally 
apply  in  this  case.  It  may  be  remarked  that  these  construc- 
tions hold  in  all  cases,  whatever  be  the  directions  of  curvature 
of  the  curves. 

The  case  where  the  moving  curve  ax  bx  is  a  right  line  is 
worthy  of  especial  notice. 


358 


Roulettes. 


In  this  case  the  normal  Om  is  perpendicular  to  the  moving 
line ;  and,  since  the  point  Pi  is  infinitely 
distant,  we  have 


COS0  I 

~oK=  oa, 


+  ok  =  ok(Art-29°); 


Fig.  67. 


whence,  P3  is  situated  on  the  lower  circle  of 
inflexions.  Hence  we  infer  that  the  dif- 
ferent centres  of  curvature  of  the  curves  en- 
veloped by  all  carried  right  lines,  at  any 
instant,  lie  on  the  circumference  of  a  circle. 

As  an  example,  suppose  the  right  line  OM  to  roll  on  a 
fixed  circle,  whose  centre   is  C2,  to 
find  the  envelope  of  any  carried  right 
line,  LM. 

In  this  case  the  centre  of  cur- 
vature, P3,  of  the  envelope  of  LM, 
lies,  by  the  preceding,  on  the  circle 
described  on  OC  as  diameter;  and, 
accordingly,  CP2  is  perpendicular 
to  the  normal  PiP2. 

Hence,  since  L  OLPx  remains 
constant  during  the  motion,  the  line 
CP2  is  of  constant  length ;  and,  if 
we  describe  a  circle  with  C  as  centre, 
and  CP2  as  radius,  the  envelope  of 
the  moving  line  LM  will,  in  all  positions,  be  an  involute  of  a 
circle.  The  same  reasoning  applies  to  any  other  moving 
right  line. 

We  shall  conclude  with  the  statement  of  one  or  two  other 
important  particular  cases  of  the  general  principle  of  this 
Article. 

(1).  If  the  envelope  a%b2  of  the  moving  curve  axbx  be  a  right 
line,  the  centre  of  curvature  Px  lies  on  the  corresponding  circle  of 
inflexions. 

(2).  If  the  moving  right  line  always  passes  through  a  fixed 
point,  that  point  lies  on  the  circle  OD2E2. 

2*92  (a).  Expression  for  Radius  of  Curvature  of 
Envelope  of  a  Right  Line. — The  following  expression 
for  the  radius  of  curvature  of  the  envelope  of  a  moving  right 


Fig.  68. 


On  the  Motion  of  a  Plane  Figure  in  its  Plane.  359 

line  is  sometimes  useful.  Let  p  be  the  perpendicular  distance 
of  the  moving  line,  in  any  position,  from  a  fixed  point  in  the 
plane,  and  w  the  angle  that  this  perpendicular  makes  with  a 
fixed  line  in  the  plane,  and  p  the  radius  of  curvature  of  the 
envelope  at  the  point  of  contact;  then,  by  Art.  206,  we  have 

p  =  P+—-  (28) 

Whenever  the  conditions  of  the  problem  give^>  in  terms  of 
u)  (the  angle  through  which  the  figure  has  turned),  the  value 
of  p  can  be  found  from  this  equation.  For  example,  the  re- 
sult established  in  last  Article  (see  Fig.  68)  can  be  easily 
deduced  from  (28).     This  is  left  as  an  exercise  for  the  student. 

293.  ©n  the  Motion  of  a  Plane  Figure  in  its  Plane. 
— We  shall  now  proceed  to  the  consideration  of  a  general 
method,  due  to  Chasles,  which  is  of  fundamental  importance 
in  the  treatment  of  roulettes,  as  also  in  the  general  investi- 
gation of  the  motion  of  a  rigid  body. 

We  shall  commence  with  the  following  theorem  : — 

When  an  invariable  plane  figure  moves  in  its  plane,  it  can 
be  brought  from  any  one  position  to  any  other  by  a  single  rotation 
round  a  fixed  point  in  its  plane. 

For,  let  A  and  B  be  two  points  of  the  figure  in  its  first 
position,  and  Ax,  Bx  their  new 
positions  after  a  displacement. 
Join  AAi  and  BBX,  and  sup- 
pose the  perpendiculars  drawn 
at  the  middle  points  of  AAX 
and  BBX  to  intersect  at  0 ; 
then  we  have  AO  =  AxO,  and 
BO  =  ByO.  Also,  since  the 
triangles  AOB  and  AxOBx 
have    their     sides    respectively  1S"    9' 

equal,  we  have  A  AOB  =  lAxOBx  ;  .\  /.  AOAx  =  lBOBx. 

Accordingly,  AB  will  be  brought  to  the  position  AXBX  by 
a  rotation  through  the  angle  AOAx  round  0.  Consequently, 
any  point  C  in  the  plane,  which  is  rigidly  connected  with  AB, 
will  be  brought  from  its  original  to  its  new  position,  Cl9  by 
the  same  rotation. 

This  latter  result  can  also  be  proved  otherwise  thus : — Join 
OC  and  0CX ;  then  the  triangles  OAC  and  OAxCx  are  equal, 


360  Roulettes. 

"because  OA  =  0AX,  AC  =  Axd,  and  the  angle  OAC,  being 
the  difference  between  OAB  and  BAC,  is  equal  to  OAxCh 
the  difference  between  0AXBX  and  BXAXCX  ;  therefore  0(7 
=  00l5  and  lAOC=Z.A1OC1;  and  hence  ziOi!  =  L  COd. 
Consequently  the  point  C  is  brought  to  Cx  by  a  rotation 
round  0  through  the  same  angle  A  OAx.  The  same  reasoning 
applies  to  any  other  point  invariably  connected  with  A  and  B. 

The  preceding  construction  re- 
quires modification  when  the  lines 
AAX  and  BBX  are  parallel.  In  this 
case  the  point,  0,  of  intersection  of  the 
lines  BA  and  BXAX  is  easily  seen  to  be 
the  point  of  instantaneous  rotation. 

For,  since  AB  =  AxBXy  and  AAX,  Fi    ?0 

BBX,  are  parallel,  we  have  OA  =  0AX, 

and  05  =  Oi?i.  Hence,  the  figure  will  be  brought  from  its 
old  to  its  new  position  by  a  rotation  around  0  through  the 
angle  AOAx. 

Next,  let  AAX)  and  BBX  be  both  equal  and  parallel.  In 
this  case  the  point  0  is  at  an  infinite  distance ;  but  it  is 
obvious  that  each  point  in  the  plane  moves  through  the  same 
distance,  equal  and  parallel  to  AAX ;  and  the  motion  is  one  of 
simple  translation,  without  any  rotation. 

In  general  if  we  suppose  the  two  positions  of  the  moving 
figure  to  be  indefinitely  near  each  other,  then  the  line  AAX, 
joining  two  infinitely  near  positions  of  the  same  point  of  the 
figure,  becomes  an  element  of  the  curve  described  by  that  point, 
and  the  line  OA  becomes  at  the  same  time  a  normal  to  the  curve. 
Hence,  the  normals  to  the  paths  described  by  all  the  points  of  the 
moving  figure  pass  through  0,  which  point  is  called  the  instan- 
taneous centre  of  rotation. 

The  position  of  0  is  determined  whenever  the  directions  of 
motion  of  any  two  points  of  the  moving  figure  are  known  ;  for  it 
is  the  intersection  of  the  normals  to  the  curves  described  by 
those  points. 

This  furnishes  a  geometrical  method  of  drawing  tangents 
to  many  curves,  as  was  observed  by  Chasles.* 

*  This  method  is  given  by  Chasles  as  a  generalization  of  the  method  of  Des- 
cartes (Art.  273,  note).  It  is  itself  a  particular  case  of  a  more  general  principle 
concerning  homologous  figures.  See  Chasles,  Sistoirc  de  la  Geometrie,  pp.  54S-9  : 
also  Bulletin  Universel  des  Sciences,  1830. 


Chasles'  Method  of  drawing  Normals.  361 

The  following  case  is  deserving  of  special  consideration : — 
A  right  line  always  passes  through  a  fixed 
point,  while  one  of  its  points  moves  along  a 
fixed  line  :  to  find  the  instantaneous  centre  of 
rotation.  Let  A  be  the  fixed  point,  and  AB  0 
any  position  of  the  moving  line,  and  take 
B 'A!  =  BA  ;  then  the  centre  of  rotation,  0,  is 
found  as  before,  and  is  such  that  OA  =  OA', 
and  OB  =  OB'.  Accordingly,  in  the  limit  the 
centre  of  instantaneous  rotation  is  the  inter- 
section of  BO  drawn  perpendicular  to  the  fixed 
line,  and  AO  drawn  perpendicular  to  the  moving  line  at  the 
fixed  point. 

In  general,  if  ABhe  any  moving  curve,  and  LM any  fixed 
curve,  the  instantaneous  centre  of  rotation  is  the  point  of  inter- 
section of  the  normals  to  the  fixed  and  to  the  moving  curves,  for 
any  position. 

Also  the  normal  to  the  curve  described  by  any  point  in- 
variably connected  with  AB  is  obtained  by  joining  the  point 
to  0,  the  instantaneous  centre. 

More  generally,  if  a  moving  curve  always  touches  a  fixed 
curve  A,  while  one  point  on  the  moving  curve  moves  along  a 
second  fixed  curve  P,  the  instantaneous  centre  is  the  point  of 
intersection  of  the  normals  to  A  and  B  at  the  corresponding 
points;  and  the  line  joining  this  centre  to  any  describing 
point  is  normal  to  the  path  which  it  describes. 

We  shall  illustrate  this  method  of  drawing  tangents  by 
applying  it  to  the  conchoid  and  the  limacon. 

294.  Application  to  Curves. — In  the  Conchoid  (Fig.  49, 
Page  33 2) ?  regarding  AP  as  a  moving  right  line,  the 
instantaneous  centre  0  is  the  point  of  intersection  of  AO 
drawn  perpendicular  to  AP,  with  BO  drawn  perpendicular  to 
LM;  and  consequently,  OP  and  OPx  are  the  normals  at  P 
and  Pi,  respectively. 

For  the  same  reason,  the  normal  to  the  Limacon  (Fig.  48, 
page  331)  at  any  point  Pis  got  by  drawing  OQ  perpendicular 
to  OP  to  meet  the  circle  in  Q,  and  joining  PQ. 


362  Roulettes. 


Examples. 

1.  If  the  radius  vector,  OP,  drawn  from  the  origin  to  any  point  P  on  a  curve, 
be  produced  to  Pi,  until  PPi  be  a  constant  length  ;  prove  that  the  normal  at  Pi 
to  the  locus  of  Pi,  the  normal  at  P  to  the  original  curve,  and  the  perpendicular 
at  the  origin  to  the  line  OP,  all  pass  through  the  same  point. 

2.  If  a  constant  length  measured  from  the  curve  be  taken  on  the  normals 
along  a  given  curve,  prove  that  these  lines  are  also  normals  to  the  new  curve 
which  is  the  locus  of  their  extremities. 

3.  An  angle  of  constant  magnitude  moves  in  such  a  manner  that  its  sides 
constantly  touch  a  given  plane  curve ;  prove  that  the  normal  to  the  curve  de- 
scribed by  its  vertex,  P,  is  got  by  joining  Pto  the  centre  of  the  circle  passing 
through  P  and  the  points  in  which  the  sides  of  the  moveable  angle  touch  the 
given  curve. 

4.  If  on  the  tangent  at  each  point  on  a  curve  a  constant  length  measured 
from  the  point  of  contact  be  taken,  prove  that  the  normal  to  the  locus  of  the 
points  so  found  passes  through  the  centre  of  curvature  of  the  proposed  curve. 

5.  In  general,  if  through  each  point  of  a  curve  a  line  of  given  length  be 
drawn  making  a  constant  angle  with  the  normal,  the  normal  to  the  curve  locus 
of  the  extremities  of  this  line  passes  through  the  centre  of  curvature  of  the  pro- 
posed. 

295.  Motion    of    any    Plane    Fignre    reduced    to 

Roulettes. — Again,  the  most  general  motion  of  any  figure 
in  its  plane  may  be  regarded  as  consisting  of  a  number  of 
infinitely  small  rotations  about  the  different  instantaneous 
centres  taken  in  succession. 

Let  0,  0',  0",  0'",  &c,  represent  the  successive  centres  of 
rotation,  and  consider  the  instant  when  /  --t 

the  figure  turns  through  the  angle  0X00'  o-  /"'''     3 

round  the  point  0.     This  rotation  will  0    ,/>-''  T2 

bring  a  certain  point  Oi  of  the  figure  to         9i*sz.~ 
coincide  with  the  next  centre  0'.  The  next    ""^^^  T^ 

rotation  takes  place  around  0';  and  suppose         o  \>-...  ^ 
the  point  Oz  brought  to  coincide  with  the  o\ 

centre  of  rotation  0' ' .  In  like  manner,  by  n'V  t' 

a  third  rotation  the  point  03  is  brought  to  1  \ 

coincide  with   0"',  and  so  on.     By  this  '    >T  » 

means  the  motion  of  the  moveable  figure  F-      2 

is  equivalent  to  the  rolling  of  the  polygon 
00i020z  .  .  .  invariably  connected  with  the  figure,  on  the 
polygon  00'0"0'"  .  .  .  fixed  in  the  plane.     In  the  limit,  the 
polygons  change  into  curves,  of   which  one  rolls,   without 


Epicyclics. 


363 


Fig.  73- 


sliding,  on  the  other ;  and  hence  we  conclude  that  the  general 
movement  of  any  plane  figure  in  its  own  plane  is  equivalent  to  the 
rolling  of  one  curve  on  another  fixed  curve. 

These  curves  are  called  by  Beuleaux*  the  "  centrodes"  of 
the  moving  figures. 

For  example,  suppose  two  points  A  and  B  of  the  moving 
figure  to  slide  along  two  fixed  right 
lines  CX  and  CY;  then  the  instan- 
taneous centre  0  is  the  point  of  inter- 
section of  AO  and  BO,  drawn  perpen- 
dicular to  the  fixed  lines.  Moreover, 
as  AB  is  a  constant  length,  and  the 
angle  ACB  is  fixed,  the  length  CO  is 
constant  ;  consequently  the  locus  of 
the  instantaneous  centre  is  the  circle 
described  with  C  as  centre,  and  CO  as 
radius.  Again,  if  we  describe  a  circle  round  CBOA,  this 
circle  is  invariably  connected  with  the  line  AB,  and  moves 
with  it.  Hence  the  motion  of  any  figure  invariably  connected 
with  AB  is  equivalent  to  the  rolling  of  a  circle  inside  another 
of  double  its  radius  (see  Art.  285). 

Again,  if  we  consider  the  angle  XCY  to  move  so  that  its 
legs  pass  through  the  fixed  points  A  and  B,  respectively  ;  then 
the  instantaneous  centre  0  is  determined  as  before.  More- 
over, the  circle  BOA  becomes  &  fixed  circle,  along  which  the 
instantaneous  centre  0  moves.  Also,  since  CO  is  of  constant 
length,  the  outer  circle  becomes  in  this  case  the  rolling  curve. 
Hence  the  motion  of  any  figure  invariably  connected  with  the 
moving  lines  CX  and  CY  is  equivalent  to  the  rolling  of  the 
outer  circle  on  the  inner  (compare  Art.  286). 

295  (a).  Epicyclics. — As  a  further  example,  suppose  one 
point  in  a  plane  area  to  move  uniformly  along  the  circum- 
ference of  a  fixed  circle,  while  the  area  revolves  with  a  uniform 
angular  motion  around  the  point,  to  find  the  position  of  the 
"  centrodes." 

The  directions  of  motion  are  indicated  by  the  arrow 
heads.     Let  C  be  the  centre  of  the  fixed  circle,  P  the  position 


*  See    Kennedy's    translation  of    Reuleaux's    Kinematics  of  Machinery, 
pp.  65,  &c. 


364 


Roulettes. 


of  the  moving  point  at  any  instant,  Q  a  point  in  the  moving 

figure   such  that   CP  =  PQ. 

Now,  to  find  the  position  of 

the   instantaneous    centre    of 

rotations   it   is    necessary   to 

get  the  direction  of  motion  of 

the  point  Q. 

Let  Pi  represent  a  con- 
secutive position  of  P,  then 
the  simultaneous  position  of  Q 
is  got  by  first  supposing  it  to 
move  through  the  infinitely 
small  length  QR,  equal  and 
parallel  to  PPi,  and  then  to 
turn  round  Px,  through  the 
angle  RPxQi,  which  the  area 
turns  through  while  P  moves 
to   Pi.     Moreover,  by  hypo- 


Fig.  74- 


thesis,  the  angles  PCPi  and  RPiQi  are  in  a  constant  ratio : 
if  this  ratio  be  denoted  by  m,  we  have  (since  PQ  =  PC) 

RQX  =  mPP1  =  mQR. 

Join  Q  and  Ql9  then  QQX  represents  the  direction  of  mo- 
tion of  Q.  Hence  the  right  line  QO,  drawn  perpendicular 
to  QQi,  intersects  CP  in  the  instantaneous  centre  of  rotation. 

Again,  since  the  directions  of  PO,  PQ,  and  QO  are,  re- 
spectively, perpendicular  to  QR,  RQi,  and  QQi,  the  triangles 
QPO  and  QiRQ  are  similar; 

.-.  PQ  =  mPO,  i.e.  CP  =  mPO. 

Accordingly,  the  instantaneous  centre  of  rotation  is  got 
by  cutting  off 

(29) 


m 


Hence,  if  we  describe  two  circles,  one  with  centre  C  and 
radius  CO,  the  other  with  centre  P  and  radius  PO;  these 
circles  are  the  required  centrodes ;  and  the  motion  is  equivalent 
to  the  rolling  of  the  outer  circle  on  the  inner. 


Epicyclics. 


365 


Accordingly,  any  point  on  the  circumference  of  the  outer 
circle  describes  an  epicycloid,  and  any  point  not  on  this  cir- 
cumference describes  an  epitrochoid.  When  the  angular 
motion  of  PQ  is  less  than  that  of  CP,  i.e.  when  m<  1, 
the  point  0  lies  in  PC  produced.  Accordingly,  in  this 
case,  the  fixed  circle  lies  inside  the  rolling  circle ;  and  the 
curves  traced  by  any  point  are  still  either  epitrochoids  or  epi- 
cycloids. 

In  the  preceding  we  have  supposed  that  the  angular 
rotations  take  place  in  the  same  direction.  If  we  suppose  them 
to  be  in  opposite  directions,  the  construction  has  to  be  modified, 
as  in  the  accompanying  figure. 

In   this   case,    the    angle  Er%p?% 

RPxQx  must  be  measured  in 
an  opposite  direction  to  that 
of  PCPi ;  and,  proceeding  as 
in  the  former  case,  the  direc- 
tion of  motion  of  Q  is  repre- 
sented by  QQi;  accordingly, 
the  perpendicular  QO  will  in- 
tersect CP  produced,  and,  as 
before,  we  have 


PO  = 


PC 


m 


Fig.  75. 


Hence  the  motion  is  equi- 
valent to  the  rolling  of  a  circle 

©f  radius  PO  on  the  inside  of  a  fixed  circle,  whose  radius  is 
CO.  Accordingly,  in  this  case,  the  path  described  by  any 
point  in  the  moving  area  not  on  the  circumference  of  the 
rolling  circle  is  a  hypotrochoid. 

Also,  from  Art.  291,  it  is  plain  that  the  envelope  of  any 
right  line  which  passes  through  the  point  P  in  the  moving 
area  is  an  epicycloid  in  the  former  case,  and  a  hypocycloid 
in  the  latter. 

Again,  if  we  suppose  the  point  P,  instead  of  moving  in  a 
circle,  to  move  uniformly  in  a  right  line,  the  path  of  any 
point  in  the  moving  area  becomes  either  a  trochoid  or  a 
cycloid. 

Curves  traced  as  above,  that  is,  by  a  point  which  moves 


366  Roulettes. 

uniformly  round  the  circumference  of  a  circle,  whose  centre  moves 
uniformly  on  the  circumference  of  a  fixed  circle  in  the  same 
plane,  are  called  epicyclics,  and  were  invented  by  Ptolemy 
(about  a.d.  140)  for  the  purpose  of  explaining  the  planetary 
motions.  In  this  system*  the  fixed  circle  is  called  the  deferent, 
and  that  in  which  the  tracing  point  moves  is  called  the 
epicycle.  The  motion  in  the  fixed  circle  may  be  supposed  in 
all  cases  to  take  place  in  the  same  direction  around  C,  that 
indicated  by  the  arrows  in  our  figures.  Such  motion  is  called 
direct.  The  case  for  which  the  motion  in  the  epicycle  is  direct 
is  exhibited  in  Fig.  74. 

Angular  motion  in  the  reverse  direction  is  called  retro- 
grade. This  case  is  exhibited  in  Fig.  75.  The  corresponding 
epicyclics  are  called  by  Ptolemy  direct  and  retrograde  epicy- 
clics. 

The  preceding  investigation  shows  that  every  direct  epi- 
cyclic  is  an  epitrochoid,  and  every  retrograde  epicyclic  a 
hypotrochoid. 

It  is  obvious  that  the  greatest  distance  in  an  epicyclic 
from  the  centre  C  is  equal  to  the  sum  of  the  radii  of  the  circles, 
and  the  least  to  their  difference.  Such  points  on  the  epicyclic 
are  called  apocentres  and  pericentres,  respectively. 

Again,  if  a  represent  the  radius  of  the  fixed  circle  or 
deferent,  and  j3  the  radius  of  the  revolving  circle  or  epicycle  ; 
then,  if  the  curve  be  referred  to  rectangular  axes,  that  of  x 
passing  through  an  apocentre,  it  is  easily  seen  that  we  have 
for  a  direct  epicyclic 

x  =  a  cos  0  +  |3  cos  mO, ) 
y  =  a  sin  t)  +  p  sin  mu.  ) 


*  The  importance  of  the  epicyclic  method  of  Ptolemy,  in  representing  ap- 
proximately the  planetary  paths  relative  to  the  earth  at  rest,  has  recently  heen 
Drought  prominently  forward  hy  Mr.  Proctor,  to  whose  work  on  the  Geometry  of 
Cycloids  the  student  is  referred  for  fuller  information  on  the  subject. 

We  owe  also  to  Mr.  Proctor  the  remark  that  the  invention  of  cycloids,  epi- 
cycloids, and  epitrochoids,  is  properly  attributable  to  Ptolemy  and  the  ancient 
astronomers,  who,  in  their  treatment  of  epicyclics,  first  investigated  some  of 
the  properties  of  such  curves.  It  may,  however,  be  doubted  if  Ptolemy  had 
any  idea  of  the  shape  of  an  epicyclic,  as  no  trace  01  such  is  to  be  found  in  the  entire 
of  his  great  work,  The  Almagest. 


Example  on  the  Construction  of  Circle  of  Inflexions.     367 

The  formulae  for  a  retrograde  epicyclic  are  obtained  by 
changing  the  sign  of  m  (compare  Art.  284). 

It  is  easily  seen  that  every  epicyclic  admits  of  a  twofold 
generation. 

For,  if  we  make  mO  =  <j>,  equation  (30)  may  be  written 

0 
x  -  B  cos  0  +  a  cos  — , 
'         r  m 

_    .  .     d> 

y  =  B  sm  0  +  a  sin  — , 

which  is  equivalent  to  an  interchange  of  the  radii  of  the 
deferent  circle  and  of  the  epicycle,  and  an  alteration  of  m 

into  — .     This  result  can  also  be  seen  immediately  geometri- 
m 

cally. 

It  may  be  remarked  that  this  contains  Euler's  theorem 

(Art.  280)  under  it  as  a  particular  case. 

296.  Properties    of  tlie  Circle    of  Inflexions. — It 

should  be  especially  observed  that  the  results  established  in 
Art.  290,  relative  to  the  circle  of  inflexions,  hold  in  all  cases 
of  the  motion  of  a  figure  in  its  plane,  and  hence  we  infer 
that  the  distances  of  any  moving  point  from  the  centre  of  curva- 
ture of  its  path,  from  the  instantaneous  centre  of  rotation,  and 
from  the  circle  of  inflexions,  are  in  continued  proportion. 

Again,  from  Art.  292,  we  infer  that  if  a  moveable  curve 
slide  on  a  fixed  curve,  the  distances  of  the  centre  of  curvature  of 
the  moving,  from  that  of  the  fixed  curve,  from  the  centre  of  in- 
stantaneous  rotation,  and  from  the  circle  of  inflexions,  are  in 
continued  proportion. 

The  particular  cases  mentioned  in  these  Articles  obviously 
hold  also  in  this  case,  and  admit  of  similar  enunciations. 

These  principles  are  the  key  to  the  theory  of  the  curvature 
of  the  paths  of  points  carried  by  moving  curves,  as  also  to  the 
curvature  of  the  envelopes  of  carried  curves. 

We  shall  illustrate  this  statement  by  a  few  applications. 

297.  Dxample  on  the  Construction  of  Circle  of 
Inflexions. — Suppose  two  curves  avbl  and  cxdh  invariably  con- 
nected with  a  moving  plane  figure,  always  to  touch  two  fixed 
curves  a2b2  and  c2d2,  to  find  the  centre  of  curvature  of  the  roulette 
described  by  any  point  Bx  of  the  moving  figure. 


368  Roulettes. 

The  instantaneous  circle  of  inflexions  is  easily  constructed 
in  the  following  manner  : — Let 
Pi  and  P3  be  the  centres  of  cur- 
vature for  the  point  of  contact 
m  for  the  curves  aj)i  and  a2b2, 
respectively :  and  let  Ql9  Q2,  he 
the  corresponding  points  for 
the  curves  c1dl  and  c2d%.     Take 

„  _     PiO2       ,  n  _     Q10* 

then,  by  Art.   290,  the  points  ™-      6 

JSi  and  Fi  lie  on  the  circle  of 

inflexions.     Accordingly,  the  circle  which  passes  through  0, 

Ei  and  Fly  is  the  circle  of  inflexions. 

Hence,   if  Bx0  meet   this   circle  in   Gi,   and  we   take 

R  O2 
RXR2  =  -jT7t>  ^he  point  R2  (by  the  same  theorem)  is  the 

centre  of  curvature  of  the  roulette  described  by  Pi. 

In  the  same  case,  by  a  like  construction,  the  centre  of  cur- 
vature of  the  envelope  of  any  carried  curve  can  be  found. 

The  modifications  when  any  of  the  curves  a-J)ly  a2b2,  &c, 
becomes  a  right  line,  or  reduces  to  a  single  point,  can  also  be 
readily  seen  by  aid  of  the  principles  already  established  for 
such  cases. 

298.  Theorem  of  Bobillier.* — If  two  sides  of  amoving 
triangle  always  touch  two  fixed  circles,  the  third  side  also  always 
touches  a  fixed  circle. 

Let  ABC  be  the  moving  triangle  ;  the  side  AB  touching 
at  c  a  fixed  circle  whose  centre  is  7,  and  AC  touching  at  b  a 
circle  with  centre  /3.  Then  the  instantaneous  centre  0  is  the 
point  of  intersection  of  bf5  and  cy. 

Again,  the  angle  j30y,  being  the  supplement  of  the  con- 
stant angle  BAG,  is  given;  and  consequently  the  instanta- 
neous centre  0  always  lies  on  a  fixed  circle. 

*  Cours  de  geometrie  pour  les  ecoles  des  arts  et  metiers.  See  also  Collignon, 
Traite  de  Mecanique  Cinematique,  p.  306. 

This  theorem  admits  of  a  simple  proof  by  elementary  geometry,  The  in- 
vestigation above  has  however  the  advantage  of  connecting  it  with  the  general 
theory  given  in  the  preceding  Articles,  as  well  as  of  leading  to  the  more  general 
theorem  stated  at  the  end  of  this  Article. 


Analytical  Demonstration. 


369 


Also  if  Oa  be  drawn  perpendicular  to  the  third  side  BC, 
a  is  the  point  in  which  the  side 
touches  its  envelope  (Art.  291). 
Produce  aO  to  meet  the  circle 
in  a ;  and  since  the  angle  aOj3 
is  equal  to  the  angle  ACB,  it 
is  constant ;  and  consequently 
the  point  a  is  a  fixed  point  on  the 
circle.  Again,  by  (4)  Art.  292, 
the  circle  j30y  passes  through 
the  centre  of  curvature  of  the 
envelope  of  any  carried  right 
line ;  and  accordingly  a  is  the 
centre  of  curvature  of  the  enve- 
lope of  BC;  but  a  has  already 
been  proved  to  be  a  fixed  point ; 
consequently  BC  in  all  positions  touches  a  fixed  circle  whose 
centre  is  a.    (Compare  Art.  286.) 

This  result  can  be  readily  extended  to  the  case  where  the 
sides  AB  and  AC  slide  on  any  curves  ;  for  we  can,  for  an  in- 
finitely small  motion,  substitute  for  the  curves  the  osculating 
circles  at  the  points  b  and  c,  and  the  construction  for  the  point 
a  will  giYe  the  centre  of  curvature  of  the  envelope  of  the 
third  side  BC. 

298  (a).  Analytical  Demonstration.— The  result  of  the 
preceding  Article  can  also  be  established  analytically,  as  was 
shown  by  Mr.  Ferrers,  in  the  following  manner  : — 

Let  a,  b,  c  represent  the  lengths  of  the  sides  of  the  moving 
triangle,  and  pl9  pi}  pz  the  perpendiculars  from  any  point 
on  the  sides  a,  b,  c,  respectively  ;  then,  by  elementary 
geometry,  we  have 

apx  +  bp2  +  cpz  =  2  (area  of  triangle)  =  2  A. 

Again,  if  pi,  /o2,  /o3  be  the  radii  of  curvature  of  the  enve- 
lopes of  the  three  sides,  and  w  the  angle  through  which  each 
of  the  perpendiculars  has  turned,  we  have  by  (28), 


api  +  bp2  +  cp3  =  2  A. 


(31) 


Hence,  if  two  of  the  radii  of  curvature  be  given  the  third 
can  be  determined. 


2  B 


37° 


Roulettes. 


We  next  proceed  to  consider  the  conchoid  of  Nicomedes. 

299.  Centre  of  Curvature  for  a  Conchoid. — Let  A 

be  the  pole,  and  LM  the  directrix  of  a  conchoid.  Construct 
the  instantaneous  centre  0,  as  before  :  and  produce  AO  until 
OAx  =  AQ. 

It  is  easily  seen  that  the  circle  circumscribing  AxORx  is 
the  instantaneous  circle  of  inflexions :  for  the  instantaneous 
centre  0  always  lies  on  this  circle  ;  also  Rx  lies  on  the  circle 
by  Art.  290,  since  it  moves  along  a  right  line  :  again,  A  lies 
on  the  lower  circle  of  inflexions  of  same  Article,  and  conse- 
quently Ai  lies  on  the  circle  of  inflexions. 

Hence,  to  find  the  centre  of  curvature  of  the  conchoid 

described  by  the  moving  point  Pi,  produce  PxO  to  meet  the 

circle  of  inflexions  in  Fu  and  take 

P  Gl 

PiP3==p!p;  then'  by  ^22)'  Fz  is 

the  centre  of  curvature  belonging  to 
the  point  Pj  on  the  conchoid. 

In  the  same  case,  the  centre  of 
curvature  of  the  curve  described  by 
any  other  point  ft,  which  is  inva- 
riably connected  with  the  moving 
line,  can  be  found.  For,  if  we 
produce  ftO  to  meet  the  circle  of 
inflexions    in    ET9   and    take    Q\Q% 

Q  O* 
=  -^=- ;  then,  by  the  same  theorem, 

Q2  is  the   centre   of  curvature  re-  Fig.  78. 

quired. 

A  similar  construction  holds  in  all  other  cases. 

300.  Snnericai  l&ouiette§. — The  method  of  reasoning 
adopted  respecting  the  motion  of  a  plane  figure  in  its  plane 
is  applicable  identically  to  the  motion  of  a  curve  on  the  sur- 
face of  a  sphere,  and  leads  to  the  following  results,  amongst 
others : — 

(1).  A  spherical  curve  can  be  brought  from  any  one 
position  on  a  sphere  to  any  other  by  means  of  a  single 
rotation  around  a  diameter  of  the  sphere. 

(2).  The  elementary  motion  of  a  moveable  figure  on  a 
sphere  may  be  regarded  as  an    infinitely  small  rotation 


Motion  of  a  Rigid  Body  about  a  Fixed  Point.  371 

around  a  certain  diameter  of  the  sphere.  This  diameter  is 
called  the  instantaneous  axis  of  rotation,  and  its  points  of 
intersection  with  the  sphere  are  called  the  poles  of  rotation. 

(3).  The  great  circles  drawn,  for  any  position,  from  the 
pole  to  each  of  the  points  of  the  moving  curve  are  normals  to 
the  curves  described  by  these  points. 

(4) .  When  the  instantaneous  paths  of  any  two  points  are 
given,  the  instantaneous  poles  are  the  points  of  intersection 
of  the  great  circles  drawn  normal  to  the  paths. 

(5).  The  continuous  movement  of  a  figure  on  a  sphere 
may  be  reduced  to  the  rolling  of  a  curve  fixed  relatively  to 
the  moving  figure  on  another  curve  fixed  on  the  sphere. 
By  aid  of  these  principles  the  properties  of  spherical  roulettes* 
can  be  discussed. 

301.  Motion  of  a  Rigid  Body  about  a  Fixed 
Point. — We  shall  next  consider  the  motion  of  any  rigid 
body  around  a  fixed  point.  Suppose  a  sphere  described 
having  its  centre  at  the  fixed  point ;  its  surface  will  intersect 
the  rigid  body  in  a  spherical  curve  A,  which  will  be  carried 
with  the  body  during  its  motion.  The  elementary  motion  of 
this  curve,  by  the  preceding  Article,  is  an  infinitely  small 
rotation  around  a  diameter  of  the  sphere  ;  and  hence  the 
motion  of  the  solid  consists  in  a  rotation  around  an  instan- 
taneous axis  passing  through  the  fixed  point. 

Again,  the  continuous  motion  of  A  on  the  sphere  by  (5) 
(preceding  Article)  is  reducible  to  the  rolling  of  a  curve 
i,  connected  with  the  figure  A,  on  a  curve  A,  traced  on  the 
sphere.  But  the  rolling  of  L  on  X  is  equivalent  to  the 
rolling  of  the  cone  with  vertex  0  standing  on  Z,  on  the  cone 
with  the  same  vertex  standing  on  X.  Hence  the  most  general 
motion  of  a  rigid  body  having  a  fixed  point  is  equivalent  to 
the  roiling  of  a  conical  surface,  having  the  fixed  point  for  its 
summit,  and  appertaining  to  the  solid,  on  a  cone  fixed  in 
space,  having  the  same  vertex. 

These  results  are  of  fundamental  importance  in  the  gene- 
ral theory  of  rotation. 


*  On  the  Curvature  of  Spherical  Epicycloids,  see  Resal ;  Journal  de  lEcoh 
Poli/technique,  1858,  pp.  235,  &c. 


2B2 


372  Examples. 


Examples. 

i.  If  the  radius  of  the  generating  circle  be  one-fourth  that  of  the  fixed, 
prove  immediately  that  the  hypocycloid  becomes  the  envelope  of  a  right  line  of 
constant  length  whose  extremities  move  on  two  rectangular  lines. 

2.  Prove  that  the  evolute  of  a  cardioid  is  another  cardioid  in  which  the 
radius  of  the  generating  circle  is  one-third  of  that  for  the  original  circle. 

3.  Prove  that  the  entire  length  of  the  cardioid  is  eight  times  the  diameter  of 
its  generating  circle. 

4.  Show  that  the  points  of  inflexion  in  the  trochoid  are  given  by  the 

equation  cos  0  +  -  =  o ;  hence  find  when  they  are  real  and  when  imaginary. 
u  a 

5.  One  leg  of  a  right  angle  passes  through  a  fixed  point,  whilst  its  vertex 
slides  along  a  given  curve ;  show  that  the  problem  of  finding  the  envelope  of 
the  other  leg  of  the  right  angle  is  reducible  to  the  investigation  of  a  locus. 

6.  Show  that  the  equation  of  the  pedal  of  an  epicycloid  with  respect  to  any 
origin  is  of  the  form 

,  ..  ad  . 

r  =  (a+  20)  cos -  -  c  cos  (0  +  a). 

a  +  20 

7.  In  figure  57,  Art.  281,  show  that  the  points  C,  P'  and  Q  are  in  directum. 

8.  Prove  that  the  locus  of  the  vertex  of  an  angle  of  given  magnitude,  whose 
sides  touch  two  given  circles,  is  composed  of  two  limacons. 

9.  The  legs  of  a  given  angle  slide  on  two  given  circles :  show  that  the 
locus  of  any  carried  point  is  a  limacon,  and  the  envelope  of  any  carried  right 
line  is  a  circle. 

10.  Find  the  equation  to  the  tangent  to  the  hypocycloid  when  the  radius  of 
the  fixed  circle  is  three  times  that  of  tbe  rolling. 

Am.  x  cos  ta  +  y  sin  w  =  b  sin  3«. 

This  is  called  the  three-cusped  hypocycloid.     See  Ex.  5,  Art.  286. 

11.  Apply  the  method  of  envelopes  to  deduce  the  equation  of  the  three- 
cusped  hypocycloid. 

Substituting  for  sin  3©  its  value,  and  making  t  =  cot  a,  the  equation  of  the 
tangent  becomes 

%$  +  (y  -  3#)  t-  +  xt  +  b  +  y  =  o, 

in  which  t  is  an  arbitrary  parameter.  If  t  be  eliminated  between  this  and  its 
derived  equation  taken  with  respect  to  t,  we  shall  get  for  the  equation  of  the 
hypocycloid, 

(#2  +  ^2)2  +  !g£2  ^2  +  y8)  +  24bx°~y  -  %3  =  27K 


Examples.  373 

12.  If  two  tangents  to  a  cycloid  intersect  at  a  constant  angle,  prove  that  the 
length  of  the  portion  which  they  intercept  on  the  tangent  at  the  vertex  of  the 
cycloid  is  constant. 

13.  If  two  tangents  to  a  hypocycloid  intersect  at  a  constant  angle,  prove 
that  the  arc  which  they  intercept  on  the  circle  inscribed  in  the  hypocycloid  is  of 
constant  length. 

14.  The  vertex  of  a  right  angle  moves  along  a  right  line,  and  one  of  its  legs 
passes  through  a  fixed  point :  show  geometrically  that  the  other  leg  envelopes  a 
parabola,  having  the  fixed  point  for  focus. 

15.  One  angle  of  a  given  triangle  moves  along  a  fixed  curve,  while  the 
opposite  side  passes  through  a  fixed  point :  find,  for  any  position,  the  centre  of 
curvature  of  the  envelope  of  either  of  the  other  sides,  and  also  that  of  the  curve 
described  by  any  carried  point. 

16.  If  a  right  line  move  in  any  manner  in  a  plane,  prove  that  the  locus  of 
the  centres  of  curvature  of  the  paths  of  the  different  points  on  the  line,  at  any 
instant,  is  a  conic. — (Resal,  Journal  de  V  Ecole  Polytechnique,  1858,  p.  112). 

This,  as  well  as  the  following,  can  be  proved  without  difficulty  from  equa- 
tion (22),  p.  352. 

17.  "When  a  conic  rolls  on  any  curve,  the  locus  of  the  centres  of  curvature 
of  the  elements  described  simultaneously  by  all  the  points  on  the  conic  is  a  new 
conic,  touching  the  other  at  the  instantaneous  centre  of  rotation. — (Mannheim, 
same  Journal,  p.  179.) 

18.  An  ellipse  rolls  on  a  right  line  :  prove  that  p,  the  radius  of  curvature  of 

the  path  described  by  either  focus,  is  given  by  the  equation  -  = ;  where 

par 

r  is  the  distance  of  the  focus  from  the  point  of  contact,  and  a  is  the  semi-axis 

major. — (Mannheim,  Ibid.) 

19.  The  extremities  of  a  right  line  of  given  length  move  along  two  fixed 
right  lines :  give  a  geometrical  construction  for  the  centre  of  curvature  of  the 
envelope  in  any  position. 

20.  Prove  that  the  locus  of  the  intersection  of  tangents  to  a  cycloid  which 
intersect  at  a  constant  angle  is  a  prolate  trochoid  (La  Hire,  Mem.  de  V  Acad,  des 
Sciences,  1704). 

21.  More  generally,  prove  that  the  corresponding  locus  for  an  epicycloid  is 
an  epitrochoid,  and  for  a  hypocycloid  is  a  hypotrochoid.  (Chasles,  Hist,  de  la 
Geom.,  p.  125). 

22.  If  a  variable  circle  touch  a  given  cycloid,  and  also  touch  the  tangent  at 
the  vertex,  the  locus  of  its  centre  is  a  cycloid.  (Professor  Casey,  Phil.  Trans., 
1877.) 

23.  Being  given  three  fixed  tangents  to  a  variable  cycloid,  the  envelope  of 
the  tangent  at  its  vertex  is  a  parabola.     (Ibid.) 

24.  If  two  tangents  to  a  cycloid  cut  at  a  constant  angle,  the  locus  of  the 
centre  of  the  circle  described  about  the  triangle,  formed  by  the  two  tangents  and 
the  chord  of  contact,  is  a  right  line.     (Ibid.) 

25.  If  a  curve  (A)  be  such  that  the  radius  of  curvature  at  each  point  is  n 
times  the  normal  intercepted  between  the  point  and  a  fixed  straight  line  (B), 


374  Examples. 

then  when  the  curve  rolls  along  another  straight  line,  {B)  will  envelope  a  curve 
in  which  the  radius  of  curvature  is  n  +  i  times  the  normal. 

Thus,  when  n  =  -  2,  {A)  is  a  parabola,  and  (B)  the  directrix  ;  and  when 
the  parabola  rolls  along  a  straight  line,  its  directrix  envelopes  a  catenary  (for 
which  n  =  —  1 ),  to  which  the  straight  line  is  directrix. 

When  the  catenary  rolls  along  a  straight  line,  its  directrix  passes  through  a 
fixed  point,  for  which  n  =  o. 

When  the  point  moves  along  a  straight  line,  the  straight  line  which  it  car- 
ries with  it  envelopes  a  circle  (n  =  1),  and  (B)  is  a  diameter. 

When  the  circle  rolls  along  a  straight  line,  its  diameter  envelopes  a  cycloid 
(n  =  2),  to  which  (B)  is  the  base.  When  the  cycloid  rolls  along  a  straight  line 
its  base  envelopes  a  curve  which  is  the  involute  of  the  four-cusped  hypocycloid, 
passing  through  two  cusps,  and  is  in  figure  like  an  ellipse  whose  major  axis  is 
twice  the  minor.     (Professor  Wolstenholme.) 

The  fundamental  theorem  given  above  follows  immediately  from  equation 
(27),  P-  357- 

26.  Prove  the  following  extension  of  Bobillier's  theorem  : — If  two  sides  of  a 
moving  triangle  always  touch  the  involutes  to  two  circles,  the  third  side  will 
always  touch  the  involute  to  a  circle. 

27.  Investigate  the  conditions  of  equilibrium  of  a  heavy  body  which  rests  on 
a  fixed  rough  surface. 

In  this  case  it  is  plain  that,  in  the  position  of  equilibrium,  the  centre  of 
gravity  G  of  the  body  must  be  vertically  over  the  point  of  contact  of  the  body 
with  the  fixed  surface. 

Again,  if  we  suppose  the  body  to  receive  a  slight  displacement  by  rolling  on 
the  fixed  surface,  the  equilibrium  will  be  stable  or  unstable,  from  elementary 
mechanical  considerations,  according  as  the  new  position  of  G  is  higher  or 
lower  than  its  former  position,  i.  e.  according  as  G  is  situated  inside  or  outside 
th#  circle  of  inflexions  (Art.  290). 

Hence,  if  pi  and  p%  be  the  radii  of  curvature  for  the  corresponding  fixed  and 
rolling  curv  3,  and  h  the  distance  of  G  from  the  point  of  contact  of  the  surfaces, 

the  equilibrium  is  stable  or  unstable  according  as  h  is  <  or  >  — .    See  Walton's 

pi  +  pz 

Problems,  p.  190  ;  also,  for  a  complete  investigation  of  the  case  where  h  = 

pi  +  pz1 
Minchin's  Statics,  pp.  320-2,  2nd  Edition. 

28.  Apply  the  method  of  Art.  285  to  prove  the  following  construction  for 
the  axes  of  an  ellipse,  being  given  a  pair  of  its  conjugate  semi-diameters  OP,  OQ, 
in  magnitude  and  position.  Prom  P  draw  a  perpendicular  to  OQ,  and  on  it  take 
PD  =  PQ  ;  join  P  to  the  centre  of  the  circle  described  on  OB  as  diameter  by  a 
right  line,  and  let  it  cut  the  circumference  in  the  points  F  and  F ;  then  the  right 
lines  OH  and  OF  are  the  axes  of  the  ellipse,  in  position,  and  the  segments  PE 
and  PF  are  the  lengths  of  its  semi-axes  (Mannheim,  Now.  An.  de  Math.  1857, 
p.  188). 

29.  An  involute  to  a  circle  rolls  on  a  right  line  :  prove  that  its  centre  describes 
a  parabola. 

30.  A  cycloid  rolls  on  an  equal  cycloid,  corresponding  points  being  in  con- 
tact :  show  that  the  locus  of  the  centre  of  curvature  of  the  rolling  curve  at  the 
point  of  contact  is  a  trochoid,  whose  generating  circle  is  equal  to  that  of  either 
cycloid. 


(     375     ) 


CHAPTER  XX. 


ON   THE    CARTESIAN    OVAL. 


302.  Equation  of  Cartesian  ©val. — In  this  Chapter" 
it  is  proposed  to  give  a  short  discussion  of  the  principal  pro- 
perties of  the  Cartesian  Oval,  treated  geometrically. 

We  commence  by  writing  the  equation  of  the  curve  in  its 
usual  form,  viz., 

n  ±  fir2  =  a, 

where  n  and  r2  represent  the  distances  of  any  point  on  the 
curve  from  two  fixed  points,  or  foci,  Fl  and  F2i  while  fx  and 
a  are  constants,  of  which  we  may  assume  that  fi  is  less  than 
unity.  We  also  assume  that  a  is  greater  than  FiF2,  the  dis- 
tance between  the  fixed  points. 

It  is  easily  seen  that  the  curve  consists  of  two  ovals,  one 
lying  inside  the  other  ;  the  former  corresponding  to  the 
equation  n  +  fir2  =  a,  and 
the  latter  to  n  -  fir2  =  a. 
Now,  with  Fi  as  centre, 
and  a  as  radius,  describe  a 
circle.  Through  F2  draw 
any  chord  DF,  join  FiD 
and  FiE;  then,  if  P  be 
the  point  in  which  FiD 
meets  the  inner  oval,  we 
have 

FD  =  a-  ri  =  fj.r2  =  p.PF2, 

From  this  relation  the 
point  P  can  be  readily 
found. 


Fig,  79- 


*  This  Chapter  is  taken,  with  slight  modifications,  from  a  Paper  published 
hy  me  in  Hermathena,  No.  rv.,  p.  509. 


376  On  the  Cartesian  Oval. 

Again,  let  Q  be  the  corresponding  point  for  the  outer 
oval  rx  -  jnr2  =  a;  and  we  have,  in  like  manner,  BQ  =  fxF2Q ; 

.-.  F2Q  :  F2P  =  QD:DP; 

consequently,  F2D  bisects  the  angle  PF2Q. 

Produce  QF2  and  PF2  to  intersect  FXE,  and  let  Px  and  Qx 
be  the  points  of  intersection. 

Then,  since  the  triangles  PF2B  and  PiF2E  are  equiangular, 
we  have  PXE  =  \xPxF2 ;  and  consequently  the  point  Px  lies  on 
the  inner  oval.  In  like  manner  it  is  plain  that  Qx  lies  on 
the  outer. 

Again,  by  an  elementary  theorem  in  geometry,  we  have 

F2P  .  F2Q  =  PD.DQ  +  F2D2 ; 

.*.  (i  -  jtr)  F2P  .  F2Q  =  F2D\ 
Also,  by  similar  triangles,  we  get 

F2P  :  F2PX  =  F2D  :  F2E ; 

consequently 

( i  -  fj2)  F2Q  .  F2PX  =  F2D  .  F2E  =  const.  (2) 

Therefore  the  rectangle  under  F2Q  and  F2PX  is  constant;  a 
theorem  due  to  M.  Quetelet. 

303.  Construction  for  Third  Focus. — Next,  draw 
QF3,  making  Z.F2QF3  =  lF2FxPx  ;  then,  since  the  points  Pl9 
Fl9  Q,  Fs  lie  on  the  circumference  of  a  circle,  we  get 

FXF2 .  F2F3  =  F2Q  .  F2PX  =  const.  (3) 

Hence  the  point  Fs  is  determined. 

"We  proceed  to  show  that  F2  possesses  the  same  properties 
relative  to  the  curve  as  Fx  and  F2 ;  in  other  words,  that  F3  is 
a  third  focus* 

"For  this  purpose  it  is  convenient  to  write  the  equation  of 
the  curve  in  the  form 

mrx  ±  lr2  =  nc3,  (4) 

in  which  c3  represents  FXF2,  and  l9  m,  n  are  constants. 
It  may  be  observed  that  in  this  case  we  have  n>m>  I. 


*  This  fundamental  property  of  the  curve  was  discovered  by  Chasles.     See 
Eistoire  de  la  Ge'ometrie,  note  xxi.,  p.  352. 


Construction  for  Third  Focus.  377 

Now,  since  L  FXF3Q  =  LFXPVF2  =  L  FXPF2,  the  triangles 
F1PF2  and  FiF3Q  are  equiangular ;  but,  by  (4),  we  have 

mF.P  +  lF^P^nF.F,; 
accordingly  we  have 

mFlF3  +  IF,Q  =  nF1Q, 

or  nFxQ  -  IF3Q  =  mFxF3 ; 

i.  e.  denoting  the  distance  from  F3  by  r3  and  FXF3  by  c2, 

wri  -  lr3  =  ^c2. 

This  shows  that  the  distances  of  any  point  on  the  outer  oval 
from  Fx  and  F3  are  connected  by  an  equation  similar  in  form 
to  (4) ;  and,  consequently,  F3  is  a  third  focus  of  the  curve. 

304.  Equations  of  Curve,  relative  to  eacb  pair  of 
Foci. — In  like  manner,  since  the  triangles  FXQF2  and  FXF3P 
are  equiangular,  the  equation 

mFxQ  -  IF2Q  =  nFxF2 
gives 

mFxF3  -  IF3P  =  nFxP. 

Hence,  for  the  inner  oval,  we  have 

nrx  +  lr3  =  mc2. 

This,  combined  with  the  preceding  result,  shows  that  the  con- 
jugate ovals  of  a  Cartesian,  referred  to  its  two  extreme  foci, 
are  represented  by  the  equation 

nrx  ±  lr3  -  mc2.  (5) 

In  like  manner,  it  is  easily  seen  that  the  conjugate  ovals  re- 
ferred to  the  foci  F2  and  F3  are  comprised  under  the  equation 

nr%  -  mr3  =  +  lcx,  (6) 

where 

cx  =  F2F3. 

305.  Relation  between  the  Constants. — The  equa- 
tion connecting  the  constants  /,  m,  n  in  a  Cartesian,  which 
has  three  points  t[9  F2,  F3  for  its  foci,  can  be  readily  found. 


378  On  the  Cartesian  Oval. 

For,  if  we  substitute  in  (3),  c3  for  FXF2)  &c,  the  equation 
is  easily  reduced  to  the  form 

Pcx  +  n%  =  m2c2, 
or  PFJFt  +  m*F^Fx  +  n2FxF2  =  o,  (7) 

in  which  the  lengths  F2FS,  &c,  are  taken  with  their  proper 
signs,  viz.,  FZFX  =  -  FXF3,  &c. 

306.  Conjugate  Ovals  are  Inverse  Curves. — Next, 

since  the  four  points  F2,  P,  Q,  Fz,  lie  in  a  circle,  we  have 

FxP.FlQ  =  F1Fz.FxF3  =  const.  (8) 

Consequently  the  two  conjugate  ovals  are  inverse  to  each  other 
with  respect  to  a  circle*  whose  centre  is  Fl9  and  whose  radius 
is  a  mean  proportional  between  FXF%  and  Fx  Fz. 

It  follows  immediately  from  this,  since  F2  lies  inside  both 
ovals,  that  Fz  lies  outside  both.  It  hence  may  be  called  the 
external  focus.  This  is  on  the  supposition  that  the  constants! 
are  connected  by  the  relations  n  >  m  >  I. 

Also  we  have 

L  PFZF2  =  lPQF2  =  l  F2QxPx  =  L  F2FZPX ; 

hence  the  lines  FZP  and  FzPi  are  equally  inclined  to  the 
axis  FXF3.  Consequently,  if  P2  be  the  second  point  in  which 
the  line  F3P  meets  the  inner  oval,  it  follows,  from  the  sym- 
metry of  the  curve,   that  the  points  P2  and  Pi   are   the 


*  It  is  easily  seen  that  when  I  =  o  the  Cartesian  whose  foci  are  FXi  F2,  Fz, 
reduces  to  this  circle.  Again,  if  n  =  o,  the  Cartesian  hecomes  another  circle, 
whose  centre  is  JF3,  and  which,  as  shall  he  presently  seen,  cuts  orthogonally  the 
system  of  Cartesians  which  have  F\,  F2,  F$  for  their  foci.  These  circles  are 
called  hy  Prof.  Crofton  (Transactions,  London  Mathematical  Society,  1866),  the 
Confocal  Circles  of  the  Cartesian  system. 

f  From  the  ahove  discussion  it  will  appear,  that  if  the  general  equation  of 
a  Cartesian  he  written  \r  +  fir'  =  vc,  where  c  represents  the  distance  hetween 
the  foci ;  then  (1)  if,  of  the  constants,  A,  fi,  v,  the  greatest  he  v,  the  curve  is 
referred  to  its  two  internal  foci ;  (2)  if  v  he  intermediate  hetween  A  and  fx,  the 
curve  is  referred  to  the  two  extreme  foci  ;  (3)  if  v  be  the  least  of  the  three,  the 
curve  is  referred  to  the  external  and  middle  focus  ;  (4)  if  X  =  fi,  the  curve  is  a 
conic ;  (5)  if  v  =  A,  or  v  —  n,  the  curve  is  a  limacon ;  (6)  if  one  of  the  constants 
A,  yu,  v  vanish,  the  curve  is  a  circle. 


Construction  for  Tangent  at  any  Point.  379 

reflexions  of  each  other  with  respect  to  the  axis  FiF2,  and  the 
triangles  F1P2F2  and  FiPiF2  are  equal  in  every  respect. 
Again,  since 

z  F2PF3  =  L  F2QF3  =  L  F*FxPx  =  L  F2FXP2, 

the  four  points  P,  P2,  Fx  and  F2  lie  on  the  circumference  of  a 
circle. 

From  this  we  have 

F3P  .  F3P2  =  FdFi .  F3F2  =  constant. 

Hence,  the  rectangle  wider  the  segments,  made  by  the  inner  oval, 
on  any  transversal  from  the  external  focus,  is  constant. 

In  like  manner  it  can  be  shown  that  the  same  property 
holds  for  the  segments  made  by  the  outer  oval. 

If  we  suppose  P  and  P2  to  coincide,  the  line  FZP  becomes 
a  tangent  to  the  oval,  and  the  length  of  this  tangent  becomes 
constant,  being  a  mean  proportional  between  F3FX  and  F3F2. 

Accordingly,  the  tangents  drawn  from  the  external  focus 
to  a  system  of  triconfocai  Cartesians  are  of  equal  length. 

This  result  may  be  otherwise  stated,  as  follows  : — A  system 
of  triconfocai  Cartesians  is  cut  orthogonally  by  the  confocal  circle 
whose  centre  is  the  external  focus  of  the  system  (Prof.  Crofton). 

This  theorem  is  a  particular  case  of  another — also  due,  I 
believe,  to  Prof.  Crofton — which  shall  be  proved  subsequently, 
viz.,  that  if  two  triconfocai  Cartesians  intersect,  they  cut  each 
other  orthogonally. 

307.  Construction  for  Tangent  at  any  Point. — 
We  next  proceed  to  give  a  geometrical  method  of  drawing 
the  tangent  and  the  normal  at  any  point  on  a  Cartesian. 

Retaining  the  same  notation  as  before,  let  R  be  the  point 
in  which  the  line  F2D  meets  the  circle  which  passes  through 
the  points  P,  F2,  F3,  Q  ;  then  it  can  be  shown  that  the  lines 
PR  and  RQ  are  the  normals  at  P  and  Q  to  the  Cartesian 
oval  which  has  Fx  and  F2  for  its  internal  foci,  and  F3  for  its 
external.     For,  from  equation  (4),  we  have  for  the  outer  oval 

dn      ,  dr2 
m  -—  -  l~—  =  o. 

as        as 


38o 


On  the  Cartesian  Oval. 


Hence,  if  wx  and  w2  be  the  angles  which  the  normal  at  Q 
makes  with  QFX  and  QF2  respectively,  we  have 


m  sin  wi  =  I  sin  w2 ;  or  sin  an :  sin  W2  =  I :  m. 


Fig.  80. 

Again,  we  have  seen  at  the  commencement  that 
l:m  =  J)Q:F2Q; 
also,  by  similar  triangles, 

BQ :  RFz  =  DQ:F2Q=l:m; 

BQ :  BF2  =  sin  BQP  :  sin  BQF2 ; 

sin BQFX :  sin jKQi^2  =  l:m. 


but 
hence 


(9) 


(10) 


Consequently,  by  (9),  the  line  BQ  is  the  normal  at  Q  to  the 
outer  oval.  In  like  manner  it  follows  immediately  that  PB 
is  normal  to  the  inner  oval. 

This  theorem  is  given  by  Prof.  Crofton  in  the  following 
form : — The  arc  of  a  Cartesian  oval  makes  equal  angles  with  the 
right  line  drawn  from  the  point  to  any  focus  and  the  circular  arc 
drawn  from  it  through  the  two  other  foci. 

This  result  furnishes  an  easy  method  of  drawing  the 
tangent  at  any  point  on  a  Cartesian  whose  three  foci  are 
given. 


Confocal  Cartesians  intersect  Orthogonally.  381 

The  construction  may  be  exhibited  in  the  following 
form : — 

Let  Fx,  F2,  F3  be  the  three  foci,  and  P  the  point  in  question. 
Describe  a  circle  through  P  and  two  foci  F%  and  F3i  and  let 
Q  be  the  second  point  in  which  FiP  meets  this  circle  ;  then 
the  line  joining  P  to  R,  the  middle  point  of  the  arc  cut  off 
by  PQ,  is  the  normal. 

308.  Confocal  Cartesians  intersect  Orthogonally. 
— It  is  plain,  for  the  same  reason,  that  the  line  drawn  from 
P  to  Bi9  the  middle  point  of  the  other  segment  standing  on 
PQ,  is  normal  to  a  second  Cartesian  passing  through  P,  and 
having  the  same  three  points  as  foci — F%  and  Fd  for  its  in- 
ternal foci,  and  Fx  for  its  external. 

Hence  it  follows  that  through  any  point  two  Cartesian  ovals 
can  be  drawn  having  three  given  points — which  are  in  directum — 
for  foci. 

Also  the  two  curves  so  described  cut  orthogonally. 

Again,  if  R C  be  drawn  touching  the  circle  PRQ9  it  is 
parallel  to  PQ,  and  hence 

F%C:FXC  =  F2R  :  RB  =  F2R2 :  F2R  .  RD ; 
but  F2R  .  RB  =  RP2 ; 

.-.  F2C  :  FXC  =  F2R2  :  PR2  =  m?  :  l\  (1 1) 

Hence  the  point  C  is  fixed. 
Again 

CR  :  FXB  =  RF2:BF2  =  m2:m2-l2; 

.\  CR  =  — — -,  (12) 

which  determines  the  length  of  CR. 

Next,  since  RP  =RQ,  if  with  R  as  centre  and  RP  as 
radius  a  circle  be  described,  it  will  touch  each  of  the  ovals, 
from  what  has  been  shown  above. 

Also,  since  C  is  a  fixed  point  by  (1 1),  and  CR  a  constant 
length  by  (12),  it  follows  that  the  locus  of  the  centre  of  a  circle 
which  touches  both  branches  of  a  Cartesian  is  a  circle  (Quetelet, 
Nouv.  Mem.  de  VAcad.  Roy.  de  Brux.  1827). 


3«2 


On  the  Cartesian  Oval. 


This  construction  is  shown  in  the  following  figure,  in 
which  the  form  of  two  conjugate 
ovals,  having  the  points  Fi9  F2, 
FS9  for  foci,  is  exhibited. 

Again,  since  the  ratio  of 
F2R  to  RP  is  constant,  we  get 
the  following  theorem,  which 
is  also  due  to  M.  Quetelet : — 

A  Cartesian  oval  is  the 
envelope  of  a  circle,  whose 
centre  moves  on  the  circum- 
ference of  a  given  circle,  while 
its  radius  is  in  a  constant  ratio 
to  the  distance  of  its  centre 
from  a  given  point. 

310.  Cartesian  Oval 
struction  has  been  given 


as 


an   Envelope. — This   con- 
in  a  different  form  by  Professor 

Casey,  Transactions  Royal  Irish  Academy,  1869. 

If  a  circle  cut  a  given  circle  orthogonally,  while  its  centre 
moves  along  another  given*  circle,  its  envelope  is  a  Cartesian 
oval. 

This  follows  immediately ;  for  the  rectangle  under  FXP 
and  FXQ  is  constant  (8),  and  therefore  the  length  of  the  tan- 
gent from  Fi  to  the  circle  is  constant. 

This  result  is  given  by  Prof.  Casey  as  a  particular  case  of 
the  general  and  elegant  property  of  bicircular  quartics,  viz. :  if 
in  the  preceding  construction  the  centre  of  the  moving  circle 
describe  any  conic,  instead  of  a  circle,  its  envelope  is  a  bicir- 
cular quartic. 


*  It  is  easily  seen  that  the  three  foci  of  the  Cartesian  oval  are  :  the  centre 
of  the  orthogonal  centre,  and  the  limiting  points  of  this  and  the  other  fixed 
circle. 


Examples.  383 


Examples. 

1.  Find  the  polar  equation  of  a  Cartesian  oval  referred  to  a  focus  as  pole. 
If  the  focus  F\  be  taken  as  pole,  and  the  line  F\F%  as  prime  vector,  we  easily 
obtain,  for  the  polar  equation  of  the  curve, 

(in2  —  l2)r2  —  2cs  (inn  —  I2  cos  0)  r  +  C32  (n2  -  I2)  —  o. 

The  equations  with  respect  to  the  other  foci,  taken  as  poles,  are  obtained  by 
a  change  of  letters. 

2.  Hence  any  equation  of  the  form 

r2  —  2  (a  +  b  cos  8)  r  +  c2  =  o 

represents  a  Cartesian  oval. 

3.  Hence  deduce  Quetelet's  theorem  of  Art.  302. 

4.  If  any  chord  meet  a  Cartesian  in  four  points,  the  sum  of  their  distances 
from  any  focus  is  constant  ? 

For,  if  we  eliminate  6  between  the  equation  of  the  curve  and  the  equation  of 
an  arbitrary  line,  we  get  a  biquadratic  in  r,  of  which  —  4a  is  the  coefficient  of 
the  second  term. 

5.  Show  that  the  equation  of  a  Cartesian  may  in  general  be  brought  to  the 
form 

S2  =  WZ, 

where  S  represents  a  circle,  and  L  a  right  line,  and  Jc  is  a  constant. 

6.  Hence  show  that  the  curve  is  the  envelope  of  the  variable  circle 

\2kL  +  2XS  +  h2  =  o. 
Compare  Art.  309. 

7.  From  this  show  that  the  curve  has  three  foci ;  i.  e.  three  evanescent 
circles  having  double  contact  with  the  curve. 

8.  The  base  angles  of  a  variable  triangle  move  on  two  fixed  circles,  while 
the  two  sides  pass  through  the  centres  of  the  circles,  and  the  base  passes  through 
a  fixed  point  on  the  line  joining  the  centres ;  prove  that  the  locus  of  the  vertex 
is  a  Cartesian. 

9.  Prove  that  the  inverse  of  a  Cartesian  with  respect  to  any  point  is  a  bi- 
circular  quartic.     (See  Salmon,  Higher  Plane  Curves,  Arts.  280,  281.) 

10.  Prove  that  the  Cartesian 

r2  -  2  (a  +  b  cos  6)r  +  c2  =  o 

has  three  real  foci,  or  only  one  according  as  ^ 

a  -  b  is  >  or  <  c. 


(     384     ) 


CHAPTEE  XXI. 


ELIMINATION    OF    CONSTANTS   AND    FUNCTIONS. 

311.  Elimination  of  Constants. — The  process  of  dif- 
ferentiation is  often  applied  for  the  elimination  of  constants 
and  functions  from  an  equation,  so  as  to  form  differential 
equations  independent  of  the  particular  constants  and  func- 
tions employed. 

We  commence  with  the  simple  example  y1  =  ax  +  b.     By 

du 
differentiation  we  get  22/—  =  a,  a  result  independent  of  b, 

ax 

A  second  differentiation  gives 

dy\        d2y 


dx)  +ydx*~°'' 

a  differential  equation  containing  neither  a  nor  b,  and  which 
accordingly  is  satisfied  by  each  of  the  individual  equations 
which  result  from  giving  all  possible  values  to  a  and  b  in  the 
proposed. 

In  general,  let  the  proposed  equation  be  of  the  form 
f(x,  y,  a)  =  o.     By  differentiation  with  respect  to  x9  we  get 

df     dfdy 
dx     dy  dx 

The  elimination  of  a  between  this  and  the  equation/^,  y,a)  =  o 

dii 
leads  to  a  differential  equation  involving  x,  y  and  — ,  which 

ax 

holds  for  all  the  equations  got  by  varying  a  in  the  proposed. 

Again,  if  the  given  equation  in   x  and  y  contain  two 

constants,  a  and  b  ;  by  two  differentiations  with  respect  to  xy 

we  obtain  two  differential  equations,  between  which  and  the 


Examples.  385 

original,  when  the  constants  a  and  b  are  eliminated,  we  get  a 

differential  equation  containing  x,  y,  —  and  -~. 

dx  ax* 

In  general,  for  an  equation  containing  n  constants,  the 

resulting  differential  equation  contains  x,  y,  —-,  —~  .  .  .  — ?  ; 

ax  ax  ax 

arising  from  the  elimination  of  the  n  constants  between  the 
given  equation  and  the  n  equations  derived  from  it  by  suc- 
cessive differentiation. 

Examples. 
1.  Eliminate  a  from  the  equation 

y2  —  2ay  +  x2  =  a2.         Ans.  (x2  —  2y2)  I  -f- }  -  Axy  —  -  x2  \—  o. 

\dxj  dx        ' 


2.  Eliminate  a  and  /8  from  the  equation 

3        d2y 
+  p  — -  =  o. 


(y-  «)3=^(^_  j8).  Ans.  2\-^-\ 

\dxj 

3.  Eliminate  the  constants  a  and  /3  from  the  equation 


d2ti 
y  =  a  cos  «#  +  #  sin  nx.  Ans.  — -  +  n2y  =  o. 

dx2 

4.  Eliminate  a  and  b  from  the  equation 

(*  -  «)2  +  (y  -  *)»  =  o2.  ^m.  c2  =  (         V2^{  ?  ■ 

This  agrees  with  the  formula  for  the  radius  of  curvature  in  Art.  226. 

5.  Eliminate  a  and  /3  from  the  equation 

y  =  ax  cos     -  +  &).  Ans.  — £  -\ f =o. 

\x         J  dx2       x4. 


6.  Eliminate  the  constants  #0,  «i,  .  .  .  ««  from  the  equation 

y  =  0  (a?)  +  tfo#n  +  aixn~l  +  ...«„.  ^ 

7.  Eliminate  the  constants  a  and  #  from  the  equation 

y  =  ae™  +  fab*. 

8.  Eliminate  a  and  J  from  the  equation 

xy  =  #0*  4-  he~x. 

9.  Eliminate,  by  differentiation,  c,  c'  from  the  equation 


dml  y 
y  =  d>(x)  +  a0xn  +  aixn~l  +  ...«„.  .4m$.  - — £  =  <Mn+1)  (#). 

y  =  ae™  +  fie**.  Ans.  — f  -  (a  +  5)  ~  +  <%  =  o. 


7  .  d2y        dy 

xy  —  aex  +  oe~x.  Ans.  x—-z  +  2~  —  xy  =  o. 

dx*         dx 


1       ,    -1-  A^y 

y  bs  are*  +  c  #e  *.  ^3ws.  a;4  — -  =  y. 

dx2 

2  C 


386  Elimination  of  Constants  and  Functions. 

312.  Elimination  of  Transcendental  Functions. — 

The  process  of  differentiation  can  also  be  employed  for  the 
elimination  of  transcendental  functions  from  equations 
of  given  form  ;  for  example,  the  logarithmic  function 
can   be   eliminated  by   differentiation    from   the    equation 

y  =  W (h (x)>  which  gives  —  =  y)  ..  We  have  met  several 
9         5rw  &        dx     <p  (x) 

instances  of  this  process  already  ;  thus,  in  Art.  86,  we 
found  that  the  elimination  of  the  symbolic  functions,  sin  and 
sin-1,  from  the  equation  y  =  sin(m  sin-^)  leads  to  the  diffe- 
rential equation 

ox  d%y        dy 
( 1  -  or)  -7^  -  x  —  +  my  =  o. 

(JjJU  (XJU 

The  principles  involved  in  this  process  are  of  great  im- 
portance in  connexion  with  the  converse  problem — viz.,  the 
procedure  from  the  differential  equation  to  the  primitive  from 
which  it  is  derived.  This  part  of  the  subject  belongs  to  the 
Integral  Calculus  in  connexion  with  the  solution  of  differential 
equations. 

Examples. 

.  .        dy 

1.  y  =  tan-1^.  Ans.  —  = 


dx      1  +  %~ 

2.  ■y  =  cos(-|.  Ans.  x2—- =  ^/ 1  -  y2  [y  —  x —  ) . 

\x)  dx  \  dx) 

3.  Eliminate  the  exponential  and  logarithmic  functions  from  the  equation 

1. 


„= 1*  (*+#■).     ^s.g +g)2= 


4.  Eliminate  the'circular  and  exponential  functions  from    y  =  ex  sin  x. 

dy 
Here  ~-  =  e*  sin  x  +  ex  cos  x  =  y  +  ex  cos  x  ; 

dx 

d2y      dy  .  dy 

therefore  ^-f  =  -f  +  e*  cos  x  -  ex  sm  x  =  2  —  -  2y. 

dx2       dx  ax 

ex  +  e-*  .       dy 

c.  y  = .  Ans.  —=i-y2. 

•J  ex  _  e-x  dx 

.    ,-.  a        nd2y       dy 

6.  y  =  sin  (log  x).  Ans.  x2  —  +  x  —  +  y  =  o. 


Elimination  of  Arbitrary  Functions.  387 

In  the  preceding  examples  we  considered  only  the  case 
of  a  single  independent  variable;  the  differential  equations 
arrived  at  in  such  cases  are  called  ordinary  differential  equa- 
tions. 

When  our  equations  are  of  such  a  nature  as  to  admit  of 
two  or  more  independent  variables,  the  equations  derived  from 
them  by  differentiation  are  called  partial  differential  equa- 
tions. We  proceed  to  consider  some  cases  of  elimination 
which  introduce  differential  equations  of  this  class. 

313.  Elimination  of  Arbitrary  Functions. — The 
equations  hitherto  considered  contained  only  two  variables ; 
we  now  proceed  to  the  more  general  case  of  an  equation  in- 
volving three  variables,  two  of  which  accordingly  can  be 
regarded  as  independent.  We  shall  denote  the  independent 
variables  by  the  letters  x  and  y>  and  the  dependent  variable 
by  z.  It  will  also  be  found  convenient  to  adopt  the  usual 
notation,  and  to  represent  the  partial  differential  coefficients 
dz      dz      d2z       d%z  d2z 

dx      dyy     dx*'     dxdy'         dy2' 

by  the  letters  p,  q,  r,  s  and  t}  respectively. 

We  proceed  to  show  that  in  this  case  we  are  enabled  by 
differentiation  to  eliminate  functions  whose  forms  are  alto- 
gether arbitrary.     In  fact  we  have  already  met  with  examples 

of  this  process ;  for  instance,  if  z  =  xn  <p  I  -  j  we  have  seen,  in 

Art.  102,  that  in  -all  cases  we  have 

dz        dz 
x—  +  y  —-  =  nz, 
dx        dy 

whatever  be  the  form  of  the  function  <p  ;  this  function  accord- 
ingly may  be  regarded  as  completely  arbitrary  in  its  form, 
and  the  preceding  differential  equation  holds  whatever  form 
is  assigned  to  it.  This  can  also  be  shown  immediately  by 
differentiation.      Conversely,  it  can  be  established  without 

difficulty  that  xn(p  I  -)  is  the  most  general  form  of  z  which 

satisfies  the  preceding  partial  differential  equation.  This 
process,  as  in  the  case  of  ordinary  differential  equations, 
comes  under  the  province  of  the  Integral  Calculus,  and  is 

2  c  2 


388  Elimination  of  Constants  and  Functions. 

mentioned  here  merely  for  the  purpose  of  showing  the  con- 
nexion between  the  integration  of  differential  equations  and 
the  formation  of  such  equations  by  the  method  of  elimination. 
As  another  simple  example,  let  it  be  proposed  to  eliminate 
the  arbitrary  function  from  the  equation  z  =f(x2  +  y2). 

Here    p  =  ^  =  2xf(x2  +  y*),   q  =  ~  =  2yf(x2  +  f)  ; 

hence  we  get  yp  -  xq  =  o  ; 

an  equation  which  holds  for  all  values  of  z  whatever  the  form 
of  the  function  (/)  may  be. 

Examples. 

1.  z  =  <p(a%  +  bij).  Ans.  aq  =  bp. 

2.  y  -  bz  —<$>(%  —  az).  ,,      ap  +  bq  =  I. 

3.  x-a  =  (z-y)<}>l^-~-y  „      {z-a)p  +  (y-0)q  =  z-7. 

4.  <£>  (x*1  +  ym)  =  zr.  „      nxn-xq  =  mym-xp. 

5.  z*  =  xy +  <!>(-)■  „     xzp  +  yzq  =  xy. 

6.  £  +  V#2  +  «/2  +  s2  =  #1_n  <t>  (-)  -  >>     z  =P%  +  <iy  +  n  V#3  +  y2  +  s2. 

314.  Condition  that  one  Expression  is  a  Function 
of  another. — Let  z  -  <p  (#),  where  v  is  a  known  function  of 

x  and  y. 

dz       ,,  .  cfa      c?s       ,,  x  dv 

Here        ^=*WS'  ^=*w;y 

dz  dv     dz  dv  dv        dv 

therefore        dxTy-TyT^0'    mpTy-qTx  =  °- 

This  furnishes  the  condition  that  z  should  be  a  function  of 
the  quantity  represented  by  v.  Also,  denoting  2  by  V,  and 
supposing  V  and  v  to  be  two  given  explicit  functions  of  x  and 
y,  the  condition  that  Vis  a  function  ofv  is  that  the  equation 

dV dv      dV dv  ,  x 

cfc  dy     dy  dx 


Condition  that  one  Expression  is  a  Function  of  another.     389 

shall  hold  for  all  values  of  x  and  y9  i.  e.  shall  be  identically 
satisfied.     For  instance,  if 


V=  v7!-^2     \A    jf    and  v  =  xy1  -f  +  yf/i-  x\ 

x  +  y 

dVdv      dVdv  . ,     ,.    ,, 

we  get  -— —  —  =  o,  identically  ; 

°  dx  dy      ay  dx 

hence  V  is  a  function  of  v  in  this  case. 

This  can  also  be  independently  verified  ;  for,  if  x  =  sin  0, 
and  y  =  sin  (f>,  we  get 

T^    cos  0  -  cos  <b         .      6  +  6 

V=  -^-a r— 7  =  -  tan : 

sin  V  -  sm  cj>  2 

v  =  sin  0  cos  0  +  cos  0  sin  0  =  sin  (0  +  <j>) ; 

which  establishes  the  result  required. 

We  have  here  assumed  that  whenever  equation  (1)  is  satis- 
fied identically,  V  is  expressible  as  a  function  of  v  :  this  can 
be  easily  shown  as  follows  : — 

Since  Fand  v  are  supposed  to  be  given  functions  of  x  and 
y,  if  one  of  these  variables,  y,  be  eliminated  between  them  we 
can  represent  Fasa  function  of  v  and  x. 

Accordingly,  let 

V=f(x9v); 

dV     df     df  dv      dV     df  dv  \ 
dx       dx      dv  dx*     dy       dv  dy ' 


therefore 


dV  dv      dV  dv      df  dv 
dx   dy     dy   dx      dx  dy 


Hence,  since  the  left-hand  side  is  zero  by  hypothesis,  we  must 

df 
have  -7-  =  o ;  i.e.  the  function  fix,  v)  or  V  reduces  to  a  func- 

dx 

tion  of  v  simply  ;  which  establishes  the  proposition. 


39°  Elimination  of  Constants  and  Functions. 

315.  More  generally,  let  it  be  proposed  to  eliminate  the 
arbitrary  function  $  from  the  equation 

where  V  and  v  are  given  functions  of  three  variables,  x,  y, 
and  s. 

Eegarding  x  and  y  as  independent  variables,  we  get  by 
differentiation 

dV       dV       ,,  Jdv        dv\ 

ix-+p^=^v\dx+pd*y 

dV       dV      ,,  ,  (dv        dv\ 

eliminating  <j>'(v)  we  obtain 

dVdv      dVdv        fdVdv      dv  dV 
dx   dy      dy  dx        \dz    dy      dz  dy 

fdVdv     dv  dV\  ,  . 

\dx  dz      dxdz  J         ' 

a  result  independent  of  the  arbitrary  function  $. 

This  equation  can  also  be  established  as  follows  : — 
Differentiating  the  equation  V '  =  <p(v),  considering  #,  y,  z 

as  all  variables,  we  get 

dV  ,  dV  7  dV  ,  ,,  x  fdv  _  dv  ,  o#  ' 
— -  do?  +  — -  aV  +  — —  dz  =  0  (v)  —  dx  +  —  dy  +  —  dz 
dx  dy  dz  \dx         dy         dz 

Then,  since  the  form  of  0  (v)  is  perfectly  arbitrary,  this  equa- 
tion must  hold  whatever  be  the  form  of  the  function  <p'(v), 
and  hence  we  must  have 

dV  7      dV  7      dV7  -) 

-—  dx  +  — —  dy  +  —-dz  =  o, 
dx  dy  dz 

>  (3) 

dv   ,        dv   7       «#   7 

-7-  Go?  +  —  tfV  +  —  as  =  o. 

a#  dy  dz  J 


Condition  that  one  Expression  is  a  Function  of  another.    391 


Moreover,  introducing  the  condition  that  %  depends  on  x 
and  y,  we  have 

d%  =  pdx  +  qdy  ; 

consequently,  eliminating  dx,  dy,  dz  between  this  and  the 
equations  in  (3),  we  get 

dV     dV     dV 

dx  '     dy  '     dz 

dv       dv        dv 
dx'      dy*      dz 

which  agrees  with  the  result  in  (2). 


=  o: 


(4) 


Examples. 

Eliminate  the  arbitrary  functions  in  the  following  cases : — 

1.  z  =  (p(a  sin  a;  -f-  b  siny). 


dz  dz 

Am.  0  cosy- — a  cos x  —  =  o. 
dx  dy 


2. 

y 
z  =  e"<p(x-  y). 

«fe       dz      z 
dx      dy      a 

3- 

z2  =  xy  +  <p  I  -  J  . 

dz         dz       xy 
dx         dy       z 

4- 

l-l-Jl-l).. 

z      x      T  \y      x) 

ndz           dz 
dx          dy 

5- 

y2$(y)  +  x 
i-x<p  (y) ' 

fy\ 

dz         dz            1 

6. 

z  =  a  \jx2  +  y2  +  <£  (  -  J  • 

x—  +  y—  =  a'Vx2  +  y2\ 
dx         dy 

7. 

Z  =  (^  +  2/)n(|)(^2-2/2). 

dz         dz 

y—  +  x— . 

dx         dy 

8.  x2  +  y2  +  z2  =  <j>  (ax  +  by  +  cz). 


ah  x  dz       .  .dz 

Am.  (dz  —  cy)  —  +  (ex  —  az)  —  =  ay  —  bx. 

CvOC  Cvtf 


392  Elimination  of  Constants  and  Functions. 

316.  Next,  let  it  be  required  to  eliminate  the  arbitrary 
function  <p  from  the  equation 

F{x,y,z,  0(w)}  =  0, 

where  u  is  a  given  explicit  function  of  x,  y,  and  z. 

Regarding  x  and  y  as  the  independent  variables,  we  may 
differentiate  the  equation  with  respect  to  x,  and  also  with 
respect  to  y  ;  then,  since  z  is  a  function  of  x  and  y,  we  have 

d .  <b(u)        ,,  ,  fdu      du  \ 

,  r/ .  0(«)  /rfw      du 

dy         ^       \dy      dz 

hence  we  obtain  two  partial  differential  equations  involving 
x,  y,  z,  p,  q,  (j>(u),  and  (p\u).  Accordingly,  if  0  (u)  and  <j>'(u)  be 
eliminated  between  these  and  the  original  equation,  we  shall 
have  a  resulting  equation  containing  only  x,  y,  z,  p,  and  q. 

317.  Case  of  two  or  more  Arbitrary  Functions. — 
If  the  given  equation  contain  more  than  one  arbitrary  func- 
tion, we  have  to  proceed  to  partial  differentiations  of  a  higher 
degree  in  order  to  eliminate  the  functions  :  thus,  in  the  case 
of  two  arbitrary  functions,  $(11)  and  \p(v)9  the  first  differen- 
tiations with  respect  to  x  and  y  introduce  the  functions  <f>'(u) 
and  4>f(v).  It  is  plainly  impossible,  in  general,  to  eliminate 
the  four  arbitrary  functions  between  three  equations ;  we 
accordingly  must  proceed  to  form  the  three  partial  differen- 
tials of  the  second  order,  introducing  two  new  arbitrary 
functions  ^\u)  and  \l/'(v).  Here,  again,  it  is  in  general 
impossible  to  eliminate  the  six  functions  between  six  equa- 
tions, so  that  it  is  necessary  to  proceed  to  differentials  of  the 
third  order  :  in  doing  so  we  obtain  four  new  equations,  con- 
taining two  additional  functions,  <t>m(u)  and  ^/"(v).  After 
the  elimination  of  the  eight  arbitrary  functions  there  would 
remain,  in  general,  two  resulting  partial  differential  equations 
of  the  third  order. 

318.  There  is  one  case,  however,  in  which  we  can  always 
obtain  a  resulting  partial  differential  equation  of  the  second 
order — viz.,  where  the  arbitrary  functions  are  functions  of 
the  same  quantity,  u. 


Case  of  Tico  or  more  Arbitrary  Functions.  393 

Thus,  suppose  the  given  equation  of  the  form 

F{x,y,  z,  <j>(u),  \fr{u))  =  o,  (5) 

where  u  is  a  known  function  of  x,  y,  and  s. 
By  differentiation  we  get 

dF       dF     dFfdu        du\  _ 
dx        dz       du  \dx        dz) 

dF       dF     dFfdu        du\  _ 
dy         dz       du  \dy        dz) 

dF 
Eliminating  —  between  these  equations,  we  obtain 
au 

dF  du      dFdu        fdFdu      dFdu\ 
dx  dy      dy  dx        \dz  dy      dy  dz) 

(dFdu     dF  du\  ... 

+  q[ =  o.  (6) 

\dx  dz      dz  dx) 

This  equation  contains  only  the  original  functions  0  (u)y 
^  (u)9  along  with  x9  y,  z,  p,  and  q.  Again,  if  we  apply  the 
same  method  to  it,  we  can  form  a  new  partial  differential 
equation,  involving  the  same  functions  <p(u)  and  ip(u)9  along 
with  x,  y9  z,  p,  q,  r9  s,  t. 

The  elimination  of  the  unknown  funotions,  <j>  (u)  and  \p  [u)9 
between  this  last  equation  and  equations  (5)  and  (6),  leads  to 
the  required  partial  differential  equation  of  the  second  order. 
The  result  in  (6)  admits  also  of  being  arrived  at  by  the 
method  adopted  in  the  seoond  proof  of  Art.  315.  For  re- 
garding x9  y9  z9  as  all  variables,  we  get  from  (5),  on  differen- 
tiation, 

dF  7      dF  _      dF  ,      dF  (du  7      du  _      du  7  \ 

-—  dx  +  -r-  d y  +  —  dz  +  -_—    —  dx  +  —  ay  +  —  dz    =0.      (7) 

dx  dy  dz  du  \dx         dy         dz      J  x  ' 

^  dF       dF      „  .       dF    f„  x 

But  ^raW)Hu)+^¥)^{u)i 


394 


Elimination  of  Constants  and  Functions. 


and  accordingly,  since  (7)  must  hold  for  all  values  of  $'(11) 
and  \p'(u)9  we  have 


and 


dF-  dF  7       dF  . 

—  dx  +  -r-dy  +  -z—  dz 
dx  dy  dz 


o 


1 


> 


(8) 


du  ,        du  7        du  _ 

dx  +  —  dy  +  —  dz  =  o.   \ 


Eliminating  between  these  equations  and 

f/s  =  pdx  +  qdy, 

we  get  the  following  determinant : 

dF     dF     dF 
dx'     dy'     dz 

du       du       du 
dx*      dy*      dz 


=  o; 


(9) 


which  plainly  is  identical  with  (6). 

This  admits  also  of  the  following  statement :  substitute  c 
instead  of  u  in  the  proposed  equation  :  then  regarding  c  as  con- 
stant, differentiate  the  resulting  equation,  as  also  the  equation 
u  =  c  (on  the  same  hypothesis) :  on  combining  the  resulting 
equations  with 

dz  =  pdx  +  qdy9 

we  get  another  equation  connecting  0  (c)  and  \p  (c)  ;  and 
applying  the  same  method  to  it,  we  obtain  the  result,  on 
eliminating  the  arbitrary  functions  cp(c)  and  \p(c)  between 
the  original  equation  and  the  two  others  thus  arrived  at. 
These  methods  will  be  illustrated  in  the  following  ex- 
amples. 


Examples.  395 

Examples. 

1.  s  =  x<p{z)  +  yty{z). 

Here  p  =  <f>  (z)  +  {a^'(a)  +  yf  (a) }  i?, 

q  =  $(z)  +  {x<f/{z)  +  yf  («)}  0. 

Hence  -  =  -^--  =  /(z),  suppose. 

?      ^(a)     Jl" 

Applying  the  principle  of  Art.  314,  we  have 
d   (p\         d  (p\ 

or  q2r  —  2pqs  +  pH  =  O. 

Otherwise  thus  :  let  z  =  c,  and  we  get  dz  =  o,  and  <j>  (c)  dx  +  if/  (c)  %  =  o ; 
also  pdx  +  qdy  =  o  ; 

therefore  I  =  *® 

q      xp(c) 

Differentiating  again,  we  have 

qdp  —  pdq  =  O, 

or  q{rdx  +  sdy}  —  p  {sdx  +  tdy)  =  o, 

which,  combined  with  pdx  +  qdy  =  o, 

leads  to  the  same  result  as  before. 

2.  z  —  %<$>  {ax  +  by)  +  y-ty  {ax  +  by) . 

Here  p  =  <j>  {ax  +  by)  +  a  {x$'{ax  -J-  by)  +  y-t/  (a%l+  by) } , 

q  =  ${ax  +  by)  +  b  {x<p'{ax  +  by)  +  y\p' {ax  +  by) } ; 

therefore  bp  —  aq  =  b<f>  {ax  +  by)  —  aty  {ax  +  by) ; 

hence  br  -  as  =  a  {b(f>' {ax  +  by)  —  a^'{ax  +  by)}, 

Is  —  at  —b  {bcpf  {ax  +  by)  -  aty {ax  +  by)}; 

therefore  b2r  —  zabs  +  a2t  —  o. 


396  Elimination  of  Constants  and  Functions. 

Otherwise  thus  :  let  ax  +  by  =  c>  then  adx  +  bdy  =  o ;  also,  dz  =  <p  (c)  dx 
4-  ty{c)  dy}  and  dz  =  pdx  +  qdy  ;  hence 

bp  -  aq  —  b<p(c)  -  aty(c). 

Differentiating  again,  we  get 

bdp  —  adq  =  0,  or  b  (rdx  +  sdy)  —  a  (sdx  +  tdy)  =  o. 

Combining  this  with  the  equation  adx  -f-  bdy  =  o,  we  get 

b2r  —  2abs  +  aH  =  o, 
as  before. 

319.  Case    of    n    Arbitrary    Functions    of    same 

Function. — It  can  be  readily  seen  that  the  preceding 
method  is  capable  of  extension  to  the  elimination  of  any 
number  n  of  arbitrary  functions  from  an  equation,  provided 
that  they  are  all  functions  of  the  same  quantity  u. 

For  the  equation  (6)  plainly  holds  in  this  case,  and,  pro- 
ceeding as  in  the  last  Article,  we  obtain  a  series  of  equations 
(the  last  being  of  the  nth  order  of  differentiation),  each  con- 
taining the  n  arbitrary  functions  along  with  the  variables  and 
their  derived  functions.  If  the  n  functions  be  eliminated 
between  the  n  differential  equations  and  the  original  equation, 
we  obtain  a  differential  equation  of  the  nth  order  which  is 
independent  of  the  arbitrary  functions  in  question. 


Examples.  397 


Examples. 


1.  Given  y  =  eax(C  +  C'x),  prove  that 

tl  _  2a  ^  +  a2y  =  o. 

2 .  Eliminate  the  constants  from  the  equation 

_,  „  „   .  .        d2y         dy 

y  =  G\elx  cos  $x  +  Cze2x  sm  3«\  Am.  — ■  —  4  —  +  13^  =  0. 

CCOO*  Cv'JQ 

3.  Eliminate  (7  and  C  from  the  equations 

COS  yyioc 
(a),  y  =  — +  C  cos war  +  C"  sin  war, 


(b)  y  —  x  sin  war  -f  C  cos  war  +  (7'  sin  war. 

d2y  d2y 

Am.  {a)  — £+  w2y  =  cos  mar.  (5)  — —  +  w2y  =  2w  cos  war, 

v      dx2  dx2 

4.  Eliminate  the  arbitrary  functions  from  the  equation 

x^v 
e  s=  -f-  +  <b  (y  +  #ar)  +  rp  (y  —  aar).  -<4ws.  r  -  a2t  =  xy. 

6 

5.  Eliminate  the  functions  from  the  equation 

o&  eft*  if         clti 

y  —  A  cos  (a  sin-1  -  +  a).  Am.  (&  —  x2)  rr-?  —  x  —  •+■  a2y  =  o. 

y  v  a       '  x  '  dx2         dx        * 

6.  Eliminate  -4  and  a  from 

y  =  A  cos  (war  +  a).  Am.  -— •  -  w  cot  war  —  +  w2y  sin2war  =  o. 

war"  war 

7.  If  3  =  cos  aar<£  {-)  +  sin  ax-ty  f  -  J ,  prove  that 

rx2  +  2sxy  +  ty2  +  a2x2z  =  0. 

8.  If  #1,  «2,  «3  be  the  roots  of  the  equation 

z%  -\-  p\z2  +pzz  +p2  =  o, 


o 


98  Examples. 


prove  that  the  result  of  eliminating  the  exponentials  from  the  equation 
y=,Gxex    +  C%e  2    +  tf3e  3 

9.  Find  the  result  of  the  elimination  of  the  arhitrary  functions  from 

z  =  <p(x  +  ay)  +  $(%  —  ay).  Ans.  a2r  —  t  -  o. 

10.  If  2  =/  (- J  +  <j>(xy),  prove  that 

/p2y  _  yi}  ^  xp  —  yq  —  o. 

11.  If  ary  +  bery  =  cex  +  de~x,  prove  that 

L^2    U*/    dx\  L  U^v     J      ^  \^3  / ' 

12.  z  =  xn<p  I- J  +  x™y\i  l-\. 

Ans.  x2r  +  2xys  +  y2t  —  (m  +  n  —  1)  (px  +  gy)  +  m>nz  =  o. 

13.  Eliminate  the  arbitrary  functions  from  the  equation 

z  —  (p  {x  +/(«/)}•  Ans.  ps  -  qr  =  0. 

14.  If  the  substitution  of  Aeax  for  2/  satisfies  the  differential  equation  with 
constant  coefficients, 

dny  dn~1y  dy 

prove  that  a  must  be  a  root  of  the  equation 

zn  +  plZn-l  +  .  .  .  +pn-iz+pn  =  O. 

15.  Eliminate  the  constants  from  the  equation 

ax2  +  2bxy  +  cy2  +  2dx  +  2ey  +  /=~o. 

Ans.  40r3  —  45gr2s  +  qq2t  =  O, 

where  *  =  g,    j  =  g,    r  =  g,&e. 


(     399     ) 


CHAPTEE  XXII. 


CHANGE    OF    THE    INDEPENDENT    VARIABLE. 

320.  Case    of  a   Single    Independent  Variable. — We 

have  already  pointed  out  the  distinction  between  indepen- 
dent and  dependent  variables  in  the  formation  of  differen- 
tial coefficients. 

In  applications  of  the  Differential  Calculus  it  is  sometimes 
necessary  to  make  our  differential  equations  depend  on  new 
independent  variables  instead  of  those  which  had  been  origi- 
nally selected. 

To  show  how  this  transformation  is  effected  we  commence 
with  the  case  of  one  independent  variable,  and  suppose  V  to 

represent  any  function  of  x,  y,  — ,  — ,  &c.      We  proceed  to 

CIX     CIX 
(it J     ft   II 

show  how  the  expressions  for  — ,  — ,  &c,  are  transformed, 

CIX    (XX 

when,  instead  of  x,  any  function  of  x  is  taken  as  the  indepen- 
dent variable. 

Let  this  new  function  be  denoted  by  t,  and  suppose  that 

dx  d  x 

— ,  — -,  &c,  are  represented  by  w,  x\  &c,  then  in  all  cases 

we  have 

du      die  dx      .  du 
dt     dx  dt        dx9 

where  u  is  any  function  of  x ; 

5M'"S5W'  (I) 

Hence  ^  =  1%  (2) 

dx     x  dt' 


400  Change  of  the  Independent  Variable. 

,  d2y      d  fdy\       d  f\  dy\      i  d  f\  dy\ 

dx2      dx  \dxj      dx  \xdt )      x  dt  \x  dt  / 

{substituting  --j-  instead  of  u  in  (i)  j; 

x  at 

.  d2y     ..dy 

d2y     Xdt*~X~dt 
hence  —?  =  JUL 2£  .  U\ 

dx-  ~3 


xc 


Again, 


'  Vy    ~*y\  fx&y-fy 

dzy      d  [     dtf         dt  id[     df        dt 


dx%      dx  \       {xf        J      xdt\         a? 


d^u  du      d"}j 


{xf 


(4) 


and  so  on  for  differentiations  of  higher  degrees. 

If  y  be  taken  as  the  independent  variable,  we  obtain  the 
corresponding  values  by  making 


dy 

=  i* 

d2y 

o,  &c. 

dt 

dt2 

d%x 

dy 

i 

d%y 

dy1 

dx 

dx' 

dy 

dx2 

fdxV' 

\dy) 

Hence  j„  -  X">      m  -  ~  TXv*;  (5) 


fd2xY    dx  d*x 
dzy  =  3\dy*)~  dy  dy\ 
dx%  (dx 


(6) 

Kdy) 

and  so  on. 

The  preceding  results  can  also  be  arrived  at  otherwise, 

as   follows.      The   essential   distinction   of  an  independent 

variable  is,  that  its  differential  is  regarded  as  constant;  ac- 

du 
cordingly,  in  differentiating  —  when  x  is  the  independent 

ax 


Case  of  a  Single  Independent  Variable.  401 

variable  we  have  d[-f)  =  ~i~-    However,  when  x  is  no  longer 

regarded  as  the  independent  variable,  we  must  consider  the 

numerator  and  the  denominator  of  the  fraction  —  as  both 

ax 

variables,  and  by  Art.  15,  we  get 

■,(dy\      dxd2y  -dyd2x         d  fdy\      dxd2y  -  dyd2x 
\dxj  dx2  '        dx\dxj  dx3 

Differentiating  again  on  the  same  hypothesis,  we  get 

d  fd2y\      dx2d3y  -  dxdyd3y  -  $dxd2xd2y  +  3  (d2x)2dy 
dx  \dx2  J  dx5 

These  results  are  perfectly  general  whatever  function  of  x 
be  taken  as  the  independent  variable.  Their  identity  with 
the  equations  previously  arrived  at  is  manifest. 

Examples. 

1.  Being  given  that  x  =  a(0  —  sin0),  y  =  a{\  -  cos  0),  find  the  value  of 

d2y  j  -i 

Ans. 


dxz'  «(i  -  cos  0)2' 

2.  Hence  deduce  the  expression  for  the  radius  of  curvature  in  a  cycloid. 

3.  If  x-  (a  +  b)  cos  0  -  b  cos  — - —  0,  y  =  (a  +  b)  sin  $  -  b  sin  — —  0,  find 

the  value  of      -r-i  • 
dx2 


«  +  *„ 
cos  0  —  cos  — - —  0 

„  dy     b 

Here  —  =  ,  *  =  tan 

dx       .   0  +  0 

sin  — - —  0  —  sin  0 
0 


GH 


d*y  a  +  2b 


4.  Change  the  independent  variable  from  x  to  0  in  the  expression  — f ,  sup- 


posing  x  =  sin  0. 

•   ndy 

gvri  Q  t_ 

dy     _i_   dy      m  d^y_  =  J_d_   f_s__  dy\  _      1     dzy     £0 

6  dx  ~  cos  0  d0i  '  '  dx7-      cos  0  d0  \cos  0  d0J     cos20  d0%        cos30 

2  1) 


402  Change  of  the  Independent  Variable. 

5.  Transform  the  equation 

„  d2y  dy      . 

x2-^  +  ax^-  +  by  =  o 
dx1  dx 

into  another  in  which  6  is  the  independent  variable,  being  given  x  =  ee. 

dy      dy  dx        dy  _ 
Here  dl~  dxdd~Xdx' 


d   (dy\  d    I    dy\         d2y  _    0d2y        dy 

hence  d~e  \de)  =  xTx  \xdx)>  oxd¥~x  d^+xdx; 

d2y      d2y      dy 
therefore  X  d^  =  W  ~dO' 

and  the  transformed  equation  is 

d2y     ,       ,  dy 

6.  Transform  the  equation 

„  d2y  dy      a2 

st ;2  — 5  +  2x-f+    .  y  =  o 
dx2  dx     x2 

into  another  where  z  is  the  independent  variable,  being  given  x  =  -. 

z 

dy  dy 

It  is  evident  that  in  this  case  x  —  =  -  z  — ,  and 

Q/$0  ctz 


d 
dx 

(•a  ■ 

d   1 
Zdz[Z 

dy\ 

dz)' 

or 

dx* 

dy 
dx 

dy 
az 

therefore 

dx2 

dy 

+  2X-f  = 

dx 

Z  dz2' 

and  the  transformed 

equation  is 

d2y 
dz* 

ly  =  0. 

7.  Change  the  independent  variable  from  x  to  z  in  the  equation 

*<z2y      o  ■,  ! 

x*  —  +  a2 y  =  o,  where  x  =  -. 
dx2  z 


d2y      2  dy 
Am.  -Z  +  -J? +  a*y  =  o. 

dz2      z  dz 


Two  Independent  Variables.  403 

321.  Two  Independent  Variables. — We  will  next 
consider  the  prooess  of  transformation  for  two  independent 
variables,  and  commence  with  the  transformations  intro- 
duced by  changing  from  rectangular  to  polar  coordinates 
in  analytic  geometry.     In  this  case  we  have 

x  =  r  cos  9,     y  =  r  sin  0  ;  (7) 

and  therefore         r2  =  x2  +  y2,    tan  0  =  -.  (8) 

x  v  J 

Accordingly,  any  function,  V,  of  x  and  y  may  be  regarded 
as  a  function  of  r  and  0,  and  by  Art.  98  we  have 


dV_dVdx     dVdy-) 
dO   .    dx  dO     dy  dO 

dV     dV  dx     dV  dy 


(9) 


dr       dx  dr      dy  dr  J 
But,  from  (7), 

dx  n      dx  .    ~  dy      .    n      dy  ,     . 

_  =  OOS0,    _  =  -,sm0  =  -y,    _  =  sme,     M  =  x;    (.0) 

hence  we  obtain 

dV_    dV_    dV 

dO  ~Xdy      y dx'  (II) 

dV       dV       dV 
rdr-=Xdx-  +  Vdi-  (I2> 

These  transformations  are  useful  in  the  Planetary  Theory. 
Again,  we  have 

dV^dVdr     dVdd-) 
dx       dr    dx     dO  dx  I 

dV_  dV(fr      dVdO  [ 
dy       dr  dy      dO  dy  , 
2  D  2 


404  Change  of  the  Independent  Variable. 

But  from  (8)  we  have 


dr       x  n     dr       .    Q 

—  =  -  =  cos  6,    -7-  =  sm  V, 
dx      r  dy 

dO  ,ny        sin  0    dd 

—  =  -  cos2  v  —  =- 
dx 


COS0 


therefore 


dV         a 

—  =  cos  d 
dx  dr 


x"  r        dy 

dV     sin  OdV 


dd' 


dV      .    ndV     cob  ddV 
dy  dr  r     du 


(14) 
(15) 
(16) 

(17) 


The  two  latter  equations  can  also  be  derived  by  solving 

for  —  and  -7—  from  the  equations  (11)  and  (12). 
dx  dy 

d2V  d2V 

X22.    Transformation  of  — — -  and  -r-r- .  —  Since1  for- 
0  dx2  dy2 

inula  (16)  holds,  whatever  be  the  form  of  the  function  F, 

we  have 

d  .  .  n  d  /  \     sin  0  d  ,  , 


dx 


dr 


r    dd 


where  <j>  stands  for  any  function  of  x  and  y.     On  substituting 

dV 

—  instead  of  6,  this  equation  becomes 

dx 


d_(dV\=  ,  bq± 
dx\dx  )  dr 


ndV     smOdV 

cos0- — 

dr  r     dd 


sin  0  d 
~r~"dd 


'      ndV     sin  OdV 

cosd- -^ 

dr  r    du 


,nd2V     cosdsmQd2V     cosdsmddV 
=  cos2  d  — 7-t^  + 


dr2 


+ 


sin0 
r 

sin0 


COS0 


drdd 
d2V 
drdd 


dd 


'cos  OdV     smOd'V 
r     dd  r     dd2 


-  sm  6  -r— 
dr  J 

]■ 


m      ^       \.      ■'    d2F     ^2F 
Transformation  oj   -r-z- 


or 


<#2F         ,nd2JP     2sin0cos0 
=  oos20-^^  + 


cfe 


e?r 


y 


rd0~      drdd 


405 


sin20dF     sin20tf2F 

+ —  + 


r     dr  r2    dd2 


In  like  manner  we  get 


d2V      .  ,nd2V     2sin0cos0ri^F     d2Vl 
=  surd 


dy 


dr2 


r  dO       drdd] 


cos2 QdV     cos2  Od2V 
r     dr  r2    dO2 


This  result  can  be  also  readily  deduced  from  the  pre- 
ceding  by  substituting  in  it  —  0  for  0. 


If  these  equations  be  added  we  have 

d2V     d2V     d2V     idV      id2V 

_i_ _i_ -j.  — 

dx2       dy1       dr2       r  dr       r2  dO2 ' 


(18) 


d2V     d2V     d2V  A 
323.  Transformation  of  — ^  +  -3-5-  +  —7^-  to  polar 


dx2        dy' 


dz* 


Coordinates. 

Let  the  polar  transformation  be  represented  by  the  equa- 
tions 

x  =  r  sin  0  cos  0,     y  =  r  sin  $  sin  0,     2  =  r  cos  0 ; 
also,  assume      p  =  r  sin  0,  and  we  have 
x  =  p  cos  0,     ?/  =  jo  sin  0 ; 


hence,  by  (18), 


d2F     d2V     d2V     idV     1  d2F 


dx2 


+ 


+  - 


+  -= 


dy2       dp2       pdp       p2  dO2 


406  Change  of  the  Independent  Variable. 

Again,  from  the  equations 

p  =  r  sin  <p,     %  =  r  cos  0, 

we  have  in  like  manner 

d^V^  d2V_  d*V  t    \_d_V      i  d2V 
dp2    '    dz2        dr%       r  dr      r2  dtp2' 

Accordingly 

d2V     d2V     d2V     d2V      idV      i  d2V      idV      i  d2V 
dx2       dy2       dz2       dr2       p  dp       p2  dO2       p  dr       r2  d(j>* 

But  by  (17)  we  have 

dV  _    .       dV     cos  <j>dVm 
dp  dr  r     d<p' 

„  1  dV     idV     cot  6dV 

tneretore  -  -7—  =  -  -7—  +  — ir-^r-  • 

p  dp       r  dr  r*    d<j> 

Hence  we  get  finally 

d*V     d2V     d2V     d2V  1       d2Y 

dx2        dy2       dz2        dr2       r2sm2<pdd2 

1  d2V     2dV     cotd>dV  ,     x 

p2  d(p2       r  dr  r2    d$ 

324.  Remarks  on  Partial  Differ entials. — As  already 

stated  in  Art.  113,  the  student  must  be  careful  to  attach  the 

correct  meaning  to  the  partial  differential  coefficients  in  each 

case. 

dx 
Thus  in  finding  —  in  (10)  we  regard  x  as  a  function  of  r 
ar 

and  6,  and  differentiate  on  the  supposition  that  6  is  constant ; 

dr 
in  like  manner  the  value  of  —  in  (14)  is  found  on  the  suppo- 

ax 

sition  that  y  is  constant. 


Geometrical  Illus  tra  tion . 


407 


The  beginner,  accordingly,  must  not  fall  into  the  con- 
fusion of  supposing  that  in  this  case  we  have  —  x  —  =  1 . 

dx     dr 

This  caution  is  necessary,  as  even  advanced  students,  from 
not  paying  proper  attention  to  the  meanings  of  partial  de- 
rived functions,  sometimes  fall  into  the  error  referred  to. 
325.  Geometrical  Illustration. — The  following  geo- 

dr  dT 

metrical  method  of  determining  the  proper  values  of  —  and  — 

dx  dr 
under  the  preceding  hypotheses  may  assist  the  beginner 
towards  forming  correct  ideas  on  this  important  subject. 

Let  P  be  the  point  whose  coordinates  are  x  and  y ;  then 
OM  =  x,    PM  =  y,    OP  =  r, 
POX  =  9.      Now,  in   finding 

-=-,  regarding   9  as    constant, 


dr 

we  take  on  the  radius  vector 
OP  produced  a  portion  PQ 
=  Ar,  and  draw  QiV  perpen- 
dicular to  OX ;  then  Ax,  the 
corresponding  increment  in  x, 
is  represented  by  MN  or  PL ; 


therefore 


Ax      PL 

X~  =  Wn  =  cos  0> 
Ar      PQ 

dr 


or 


dx 
dr 


=  cos  0. 


Again,  to  find  —  on  the  supposition  that  y  is  constant : 

let  MN  be  Ax,  the  increment  in  x,  and  draw  the  parallelo- 
gram PLMN,  and  join  OL,  meeting  in  /  a  circle  described 
with  radius  r  and  centre  0 ;  then  LI  represents  the  corre- 
sponding increment  in  r,  and  we  have 

—  =  limit  of  —  =  limit  of  -=r—  =  cos  6, 
dx  Ax  PL 

dr  dx 

so  that  in  this  case  the  values  of  —  and  —  are  each  equal  to 


cos  9  or  -,  as  before. 
r 


dx 


dr 


408  Change  of  the  Independent  Variable. 

dr   dO 
The  values  of  — ^,  — ,  &c,  can  be  also  readily  represented 
do  dec 

geometrically  in  a  similar  manner. 

326.  Linear  Transformations. — If  we  are  given 

x  =  aX+  bY+  cZ,  y=a'X  +  b'Y+c'Z,  z=a"X+b"Y+  c"Z,    (20) 

then  any  function  V,  of  x,  y  and  s,  is  transformed  into  a  func- 
tion of  X,  Yy  Z\  and,  as  in  Ex.  2,  Art.  98,  we  have 

dV_    dV      ,dV      „d_V 
dX        dx  dy  dz 

dV_hdV    h,dV    h„dV 
dY       dx  dy  dz  ' 

dV_     dV      ,dV      „dV 
dZ        dx  dy  dz 


Again,  proceeding  to  second  differentiation,  we  get 

fdV     „dV\     ,d(dV    ,dV     „dV\ 
—  +  a  —  )+af—[a—-+a—  +  a  — 
dy  dz  J       dy\   dx         dy  dz   J 


(PV_    d_(  dV 
dX2      dx  \  dx 


„  d  (  dV      ,dV      „dV 

+  a   —  [a —  +  a  -—  +  a 


dz  \  dx         dy  dz 

^d2V         ,d2V  „d2V       ,  „d2V 

=  a"  T7  +  2aa  ~—r-  +  2aa  -=—7  +  2a  a  -r- 
dx2  dxdy  dxdz  dz& 

,2d>v    „,d>r 

Similarly  we  have 

dY2  dx2  dy2  dz2  dxdy 

+  iW -——  +2b  b   — -  ; 
dxdz  dydz 


Orthogonal  Transformations.  409 

<ZZ=^^Z    ^^Z+^JZ    zcc'  — 
dZ2  dx2  dy%  d%2  dxdy 

„d2V        ,  „d2V 

+  2CC    -—r  +  2C  C    -—- . 

dxd%  dyd% 

327.  Orthogonal  Transformations. — If  the  transfor- 
mation be  such  that 

x%  +  y2  +  z2  =  X2  +  Y2  +  Z2, 
we  have 

a2  +  d2  +  d'2  =  i,        b2+bn +  b"2=i,        c*  +  cn  +  c"*  =  1.       (21) 

ab  +  db'  +  d'b"  =  o,    ac  +  dc'  +  d'c"  =  o,    bc  +  b'c'+b"c"  =  o.     (22) 

Again,  multiplying  the  first  of  equations  (20)  by  a,  the 
second  by  a',  and  the  third  by  d\  we  get  on  addition,  by  aid 
of  (21)  and  (22), 

X  =  ax  +  dy  +  d'z. 

In  like  manner,  if  the  equations  (20)  be  respectively 
multiplied  by  b,  &',  b'\  we  get 

Y  =  bx  +  b'y  +  b"z  ; 
similarly 

Z  =  ex  +  c'y  +  c"z. 

If  these  equations  be  squared  and  added,  we  obtain 

a?  +  b2  +  c2  =  1,    a'2  +  b/2  +  c'2  =  1,     a"2  +  b"2  +  c"2  =  1 .      (23) 
ad  +  bb'  +  ccf  =  o,  m"  +  bb'f  +  cc"  =  o,  ddr  +  b'b"  +  e'e"  =  o.    (24) 

Hence  in  this  case,  if  the  equations  of  the  last  Article  be 
added,  we  shall  have 

d2V     d2V    d2V_d2V     cPV    (PV 
dx2  +  dy%  +  dz2  "  dX%  +  dY2  +  dZ2 '  ^ 


4-io  Change  of  the  Independent  Variable. 

The  transformations  in  this  and  the  preceding  Article 

are  necessary  when  the  axes  of  co-ordinates  are  changed  in 

Analytic  Greometry  of  three  dimensions ;  and  equation  (25) 

shows  that,  in  transforming  from  one  rectangular  system  to 

d2V     d2V     d2V 
another,  the  function  —rT  +  ——r  +  -TT-  is  unaltered. 

dx*       dy*        dz" 

328.  General   Case   of  Transformation  for    Two 

Independent  Variables. — Suppose  that  we  are  given  the 

equations 

*  =  *(*-,  0),     t/  =  +{r,9),  (26) 

then  any  function  V  of  x  and  y  may  be  regarded  as  a  function 
of  r  and  0,  and  we  have,  from  (9), 

dV_dVdx      dVdy 
dO  'dxdQ^d^dQ' 

dV_dVdx     dVdy 
dr       dx  dr      dy  dr9 

where  the  values  of  -^,  -^,  — ,  ~  can  be  determined  from 

da  dv  dr  dr 

equations  (26). 

Whenever  these  equations  can  be  solved  for  r  and  0, 

separately,  we  can  determine,  by  direct  differentiation,  the 

values  of  — ,  —,—,—,  and  hence  by  substituting  in  (13) 
ax  ay  ax  ay 

we  can  obtain  the  values  of  -T—  and  -7—. 

dx  dy 

When,  however,  this  process  is  impracticable  we  can  ob- 
tain the   values   of  t->  -7-,  &c.,  by  solving  for  —  and  — 

ax  ay  ax  ay 

from  the  preoeding  equations. 

Thus,  we  obtain 

dV dy     dV  dy 

dV     aWdf'~dr"d9  ,     N 

—  =  ..  •  (27) 

dx        dx  dy      dx  dy 

dO  dr      dr  dd 


Transformation  for  Two  Independent  Variables.         411 

dVdx  dV dx 
dV  dO^'aVdO 
dy       dx  dy      dxdy  ^     ' 

dr  dd     dO  dr 

d2V  d2V 
The  values  of  -r^-,  -r— -,  &c.,  can  be  deduced  from  these : 
dx2     dy2 

but  the  general  formulae  are  too  complicated  to  be  of  much 

interest  or  utility. 

329.  Concomitant  Functions. — We  add  one  or  two 

results  in  connexion  with  linear  transformations,  commencing 
with  the  case  of  two  variables.  We  suppose  x  and  y  changed 
into  aX  +  bY  and  a'X  +  b'Y,  respectively,  so  that  any  func- 
tion (j)(x,  y)  is  transformed  into  a  function  of  X  and  Y;  let 
the  latter  be  denoted  by  0i  (X,  Y),  and  we  have 

<t>{x,y)  =  <f>1(X,  Y). 

Again,  let  x'  and  y'  be  transformed  by  the  same  substitu- 
tions, i.e., 

x'  =aXf+b  Y\     y'  =  a'X'  +  V  Y'; 

then  since        x  +  kx'  =  a(X  +  kX')  +  b^Y+kY'), 

and  y  +  ky  =  d(X  +  kX')  +  b\Y+kY), 

it  is  evident  that 

cf>(x  +  kx,  y  +  ky')  =  fa(X  +  kX\   Y+kY'). 

Hence,   expanding  by  the   theorem    of   Art.   127,    and 
equating  like  powers  of  k,  we  get 

x>d±+/±=X'%+r%,  (29) 

dx     "  dy  dX  d¥  v     ; 

x^  +  2xy^Ax>^2TY'p^+Y-% 
dx2  J  dxdy    J   dy2  dX2  dXdY  dY2 

&c.  &c.  (30) 


412  Change  of  the  Independent  Variable, 

Accordingly,  if  u  represent  any  function  of  x  and  y,  the 
expressions  denoted  by 

(  ,d         d\  {  ,  d  dY 

[x  —  +y  —)ii,      [x  -r  +  y  —     u,  &c, 
V    dx    u  dyj    '      V    dx     y  dy)     '        ' 

are  unaltered  by  linear  transformation. 

Similar  results  obviously  hold  for  linear  transformations 
whatever  be  the  number  of  variables  (Salmon's  Higher 
Algebra,  Art.  125). 

Functions,  such  as  the  above,  whose  relations  to  a  quantic 
are  unaltered  by  linear  transformation,  have  been  called  con- 
comitants by  Professor  Sylvester. 

330.  Transformation  of  Coordinate  Axes. — When 
applied  to  transformation  from  one  system  of  coordinate 
axes  to  another,  the  preceding  leads  to  some  important 
results,  by  applying  Boole's  method*  (Salmon's  Conies, 
Art.  159). 

For  in  the  case  of  two  dimensions  when  the  origin  is 
unaltered  we  have 

x2  +  2x1/  cos  w  +  y2  =  X'2  +  zX'Y'  cos Q  +  Yr\     (31) 

where  w  and  £2  denote  the  angle  between  the  original  axes 
and  that  between  the  transformed  axes,  respectively. 

Multiply  (31)  by  A,  and  add  to  (30):  then  denoting 
<p  {x,  y)  by  u,  and  $i(X,  Y)  by  IT,  we  get 

rd2u      .  \         ,  ,(  d2u       .  \       ,Jdhi 

=*^H-r(^+x„.Q),r..(£?+x). 

Now,  suppose  A  assumed  so  as  to  make  the  first  side  of 
this  equation  a  perfect  square,  it  is  obvious  that  the  other 
side  will  be  a  perfect  square  also.  The  former  condition 
gives 

'—  \\(—  \\  ( d%u   x      Y 

Kdx~       )\dy2       J     \dxdy  J9 

_  *   I  am  indebted  to  Prof.  Burnside  for  the  suggestion  that  the  equations  of 
this  Article  are  immediately  obtained  by  Boole's  method. 


Transformation  of  Coordinate  Axes.  413 

....        .  fd2u     d2u         d2u 
or         A2  sin3 u)  +  X[  -^  +  -7-r  -  2  -r— -  cos  w 
\dx2      dy*        dxdy 

d2u  d2u      f  d2u  \? 
efo?2  efo/2      \dxdyj 

Accordingly,  we  must  have  at  the  same  time 

^  .  ,~     x  (^TI     d2U         d2U  \ 

X  sm  °  +  H^"2  +  dY* ~  2  ^X^F  C0Sl2 J 

d2Ud2U     f  d2TJ  \2 


+  tfX2^F2     [dXdYJ     °' 
Hence,  comparing  coefficients,  we  get 

d"ud2u  _frd2u_\2    d2Ud2U     (  d2U  V 
~dx^dy%~  \dx~dy]      dX2dY~2 ~ \dXdYJ 
sin2w  sin2Q 

and 

d2u     d2u         d2u  d2U     d*U  d2U 

d^+df-2dx^yG0Sl0     dX2+^Y2-2dXdYQ°*® 
sin2w  sin2 12 


(32) 


(33) 


Consequently,  if  u  be  any  function  of  the  coordinates  of 
a  point,  the  expressions 

d2u  d2u     f  d2u  Y         d2u     d2u         d2u 

dx2  dy2      \dxdyj       Adx2      dy2      " dxdy 

.  9  and.  .  o 

sin2  w  snr  w 

are  unaltered  when  the  axes  of  coordinates  are  changed  in  any 
manner,  the  origin  remaining  the  same. 

In  the  particular  case  of  rectangular  axes,  it  follows  that 

d2u     d2u         d2u  d2u      f  dhi  Y 
dx2      dy2  dx2  dy2      \dxdyj 

preserve  the  same  values  when  the  axes  are  turned  round 
through  any  angle. 


4J4 


Change  of  the  Independent  Variable. 


331.  Application  to   Orthogonal  Transformation. 

— It  is  easy  to  extend  the  preceding  results  to  three  or  more 
variables  when  the  transformations  are  orthogonal  (Art.  327). 
Thus,  in  the  case  of  three  variables  we  have 

x,%  +  y2  +  z2  =  X'2  +  Y'2  +  Z'\ 

Multiplying  this  by  A  and  adding  the  result  to  the  equation 
that  corresponds  to  (30),  it  follows  that  the  expression 


x 


,fd2u      .\        ,Jd2u        \        ,Jdhi        \ 


d2u 
dydz 


+  2z'x 


dhc 
dzdx 


+  2xy 


,  ,  d2u 


dxdy 


is  unaltered  by  orthogonal  transformation. 

Next,  suppose  that  A  is  such  that  the  quadratic  function 
in  x',  yr  and  z  is  the  product  of  two  linear  factors  ;  then,  by 
Art.  107,  we  have 


d2u     . 
d2u 


dxdy* 
d2u 


d2u 
dxdy* 

d2u     . 

W    ' 

d2u 


d2u 
dxdz 

d2u 


j„> 


dxdz9       dydz 


dydz 
d2u     . 


=  o. 


(34) 


But,  as  the  transformed  expression  must  also  be  the  product 
of  two  linear  factors,  we  have 


d2u     x      d2u     d2u 
— 2  +  A, 


dx2 


d2u    d2u     A 


dxdy9  dxdz 
d2u 


dydx  dy' 


dydz 


d2u     d2u    d2u     , 
dxdz9  dydz9  dz2 


d*U    . 
dX2  +  *9 


d2U     d2U 
dXdY9  dXdZ 


d2TJ    d2U     .     d2U 


dXdY9  dY2  '  '•'  dYdZ 
d2JJ     d2U    d2U 


dXdZ9  dYdZ9  dZ2 


+  A 


(35) 


Orthogonal  Transformation. 


4i5 


Equating  the  coefficients  of  like  powers  of  A,  we  see  that  the 
expressions 

d2u     d2u      d2u 
ay" 


dx2      dv2      dz2 ' 


d2u  d2u 
dec2  dy2 

and 


d2u\2    <Pud2u      ( d2u  V      d~udhi 
dxdy)     dec2  dz2      \dxdzj       dy2  dz2 


dx2  dz2      \dxdzj 

d2u 

d2u 

d2u 

dx2' 

dxdy 

dxdz 

d2u 

d2u 

d2u 

dxdy' 

df[ 

dydz 

d2u 

d2u 

dhc 

'  d2u  Y 
,  dydz  J 


dxdz'     dydz'     dz2 


are  unaltered  by  orthogonal  transformation. 

The  first  of  these  results  has  been  already  arrived  at  by 
direct  substitution  (Art.  327). 


Jacobians. 

332.  The  results  in  the  preceding  Article  are  particular 
cases  of  a  class  of  general  theorems  in  determinants,  which 
were  first  developed  by  Jacobi  (Crelle's  Journal,  1841). 

Thus,  if  u,  v,  w  be  functions  of  x,  y,  z9  the  determinant 


J  = 


du  du  du 

dx'  dy'  dz 

dv  dv  dv 

dx'  dy'  dz 

dw  dw  dw 

dx'  dy'  dz 


(36) 


was  styled  by  Jacobi  a  functional  determinant       Such  a 


416 


Jacobians. 


determinant  is  now  usually  represented  by  the   notation 

d(u9  v,  w) 
d{x9  y,  z)' 

and  is  called  the  Jacobian  of  the  system  u,  v,  w  with  respect 
to  the  variables  x9  y9  %. 

In  the  particular  case  where  u,  v,  w  are  the  partial  diffe- 
rential coefficients  of  the  same  function  of  the  variables  x9  y,  z9 
their  Jacobian  becomes  of  the  form  (35),  and  is  called  the 
Hessian  of  the  primitive  function.  Thus  the  determinant  in 
(35)  is  called  the  Hessian  of  u,  after  Hesse,  who  first  in- 
troduced such  functions  into  analysis,  and  pointed  out  their 
importance  in  the  general  theory  of  curves  and  surfaces. 

More  generally,  if  yl9  y29  y3 . . .  yn  be  functions  of  xl9  x29  x3> 
.  .  .  xn9  the  determinant 


dyi 
dxi 

<fyz 

dx^ 


dx2 
dy% 

dx2 


dyi 

ClXn 

dy2 

UiXin, 


<tyn  dyn  <fyn 

dx\        dx2        '  dxn 


is  called  the  Jacobian  of  the  system  of  functions  yl9  y%9*\.  yn 
with  respect  to  the  variables  xi9  x29  .  .  .  xn ;  and  is  denoted  by 


d(xh  X2,   .  .  .  Xn)' 


(37) 


Again,  if  yl9  y29  ...  yn  be  differential  coefficients  of  the 
same  function,  the  Jacobian  is  styled,  as  above,  the  Hessian 
of  the  function.  A  Jacobian  is  frequently  represented  by  the 
notation 

J(yi,  y*>  •  •  -  yn), 

the  variables  xl9  x%9  .  .  .  xn  being  understood. 


Jacobians.  417 

If  the  equations  for  yly  y2}  . . .  yn  be  of  the  following  form  : 

yi  =/(4 
y%  =M%i,  a*), 
y%  =/3(^i,  #2,  #3), 

it  is  obvious  that  their  Jacobian  reduces  to  its  leading  term, 
viz., 

J=dyi   dy*       dyn  .  g. 


» 


This  is  a  case  of  a  more  general  theorem,  which  will  be 
given  subsequently  (Art.  336). 


Examples. 

1.  Find  the  Jacobian  of  y\,  y%,  ...  yn,  being  given 

y\  —  1  -  »i,    ^2  =  a?i(i  -  #2),     ^3  =  #1  #2(1  -  %i)  ... 

^n=  #1  #2  • .  •  #»-i(i  -  fl?»).  ^««.  /=  (-  iJ'W1^"-2  .  .  .  %n-i> 

2.  Find  the  Jacobian  of  x\,x%,...  xn  with  respect  to  0i,  02,  •  •  .  0«,  being 
given 

#1  =  cos  0i,     #2  =  sin  0i  cos  02,     %%  =  sin  0i  sin  02  cos  03,  .  .  . 

xn  =  sin  0i  sin  02  sin  03  ...  sin  0„-i  cos  0n. 

d(xi,  X2,  .  .  •  xn)  ,     . 

■^ws-  ^/n    a   T-  =  (~  J)n  smM0i  •  sin*"1  02  .  .  .  sm0M. 

»(01j  02,    •   •    •   0«) 

333.  Case  of  tSae  Functions  not  being  Indepen- 
dent.— If  the  system  yx,  y2,  .  .  .  yn  be  connected  by  a  re- 
lation, it  is  easily  seen  that  their  Jacobian  is  always  zero. 

For,  suppose  the  equation  of  connexion  represented  by 

f&i,  y*>  •  •  •  yn)  =  o ; 

2  E 


4 1 8  Jacolians. 

then,  differentiating  with  respect  to  the  variables  xly  x2  .  .  .  xny 
we  get  the  following  system  of  equations : — 

dF  dy-i      dF  dy2  ^  dF  dyn  _ 

dyx  dxx      dy2  dxi  dyn  dxx 

dF  dyx      dF  dy2  dF  dyn 

dyx  dx2      dy2  dx2  dyn  dx2 


dF  dyx      dF  dy2  dF  dyn 

dyx  dxn      dy2  dxn  dyn  dxn 

,.    .     L.        dF     dF  dF 

whence,  eliminating    — ,     — ,  .  .  .  — ,     we  get 

dyx      ay2  ayn 

d(yi,  y%>  ■ . .  yn)  =  Q  ,    , 

The  converse  of  this  result  will  he  established  in  Art.  337  ; 
and  we  infer  that  whenever  the  Jacobian  of  a  system  of 
functions  vanishes  identically,  the  functions  are  not  indepen- 
dent. This  is  an  extension  of  the  result  arrived  at  in  Art.  314. 

334.  Case  of  Functions  of  Functions. — If  we  sup- 
pose Ui,  u2,  u3  to  be  functions  of  yly  y2i  yd,  where  y1}  y2,  yd  are 
given  functions  of  xlf  x2,  x3 ;  then  we  have 

dui      dui  dyx      dux  dy2      dux  dyz 
dxx      dyx  dxi      dy2  dxx      dyz  dxx 

dux      dux  dyx      dux  dy2     dux  dyz 

dx2      dyY  dx2      dy2  dx2      dy3  dx2 

dux      dux  dyx      dux  dy2      dux  dy3 
dx3      dyx  dx3     dy2  dx3      dy%  dx3 

&c. 


General  Theorem  on  Jacobians. 


419 


Hence,  by  the  ordinary  rule  for  the  multiplication  of  de- 
terminants, we  get 


dux      dux      dux 
dx^     dx2      dxz 

i 

du2      du2      du2 

ClXx        CtX2        dX3 
j 

\du3      du3      du3 

OjXx        Ct/0S2        0/X3 

dux      dtix      dux 
dyx      dy2      dy3 

du2      du2     du2 
dyx      dy2      dy3 

du3      du3      du3 
dy'     dy2'     dy3 

• 

dyx      dyx      dyt 

OjXx       CvX-2,       CIX3 

dy*      dy2      dy2 
dxx      dx2     dx3 

dy3      dy3      dy3 
dxx      dx2      dx3 

(40) 


or 


d(ux,  u2,  u3)  _  d(ux,  u2,  u3)      d(yx,  y2,  y3) 
d(x1}  x2,  x3)      d{yx,  y2,  y3)  '  d(xx,  x2,  x3)' 

It  follows  as  a  particular  case,  that 

d{yi>y*>  ^3)    d(xh  x2,  x3) 


d{xx,  x2,  x3)      d(yx,  y2,  y3) 


=  1. 


(41) 


These  results  are  readily  generalized,  and  it  can  be  shown 
by  the  method  given  above,  that 

d(ux,  u2,  ..  .  un)  ~  d(uu  u2,  ...  un)      d(yx,  y2i  .  . .  yn) 
d(xly  #2,  •  •  •  %n)     dt(yx,  y*,  ...  yn)  '  d(xif  x2, . . .  xn)'  ^  ' 

This  is  a  generalization    of  the    elementary  theorem 
(Art.  19), 

du      du  dy 

dx      dy  dx' 

Again, 

d(yx,  3/2,  • .  •  yn)  d(xx,  x2,  .  . .  xn)  _ 

dt(xx,  x2,  . . .  xn)  d(yl9  y2,  . . .  yn) 


(43) 


This  may  be  regarded  as  a  generalization  of  the  result 


dx       1 
dy  "  dy' 
dx 

2  E  2 


420  Jacobians. 

335.  Jacobian    of   Implicit    Functions. — Next,    if 

u,  v,  w,  instead  of  being  given  explicitly  in  terms  of  x,  y,  z} 
be  connected  with  them  by  equations  such  as 

Fi{x,y,z,u,v,w)  =  o,  F2(x,y,z,u,v,w)  =  o,  Fz(x,y,zyu9v,w)  =  o, 

then  u,  v,  w  may  be  regarded  as  implicit  functions  of  x,  y,  z. 
In  this  case  we  have,  by  differentiation, 

dFx      dFx  du     dFx  dv      dFx  dw  _ 
dx       du    dx      dv     dx     dw    dx 

dFx     dFt  du     dFx  dv      dFx  dw  _ 
dy       du    dy      dv     dy     dw    dy 


dF2     dFz  du      dF2  dv      dF2  dio 


+ 


—  + 


—  +  -r—  T-  =  O, 


dx       du     dx      dv     dx      dw    dx 


Hence  we  observe,  from  the  ordinary  rule  for  multipli- 
cation of  determinants,  that 


du 

dv 

dw 

dx' 

dx* 

dx 

du 

dv 

dw 

dy' 

W 

dy 

du 

dv 

dw 

dz' 

Tz* 

dz 

(44) 


dFx    dFi    dFx 
du  '   dv  '   dw 

dll    dF*    dF, 
du  '    dv  '    dw 

dF3    dFs    dF* 
du  '   dv  '   dw 

This  result  may  be  writtten 

d{Fx,  -Fa,  F3)      d(u,  v,  w) 
d{u,v,tv)      'd(x,y,z)  d(x,yfz) 

The  preceding  can  be  generalized,  and  it  can  be  readily 
shown  by  a  like  demonstration  that  if  yi9  yif  y^  •  •  •  yn 


dF\  dFx  dFx 

dx  '  dy  '  dz 

dF%  dF\  dl<\ 

dx  '  dy  '  dz 

dF\  dFs  dJF\ 

dx  '  dy  '  dz 


d(Fx,  F„  Fz) 


Jacobian  of  Implicit  Functions.  421 

are  connected  with  xi9  x2,  xz  . . .  xn  by  n  equations  of  the 
form 

Fl  (a?!,  x2  . .  .  xn9     yl9  y2  . . .  yn)  -  o, 

F2  (xh  x2  ...  xn,     yl9  y2  ..  .  yn)  =  o, 


Fn  (xl9  x2  . ..  xn,     yl9  y2  .  . .  yn)  =  o, 

we  shall  have  the  following  relation  between  the  Jacobians  : 

d(Fl9  F29...  Fn)     d{yxy29...yn)  d{Fl9F29...Fn) 

d(yi>  y*>  •  •  •  yn)    '  d[x9  x29  ...  xn)  d(xl9  x29.  . .  xn)' 

Accordingly 

d(Fl9  F29...  Fn) 
d{yi,  y2, . .  .yn)     ,     ,n  d(xl9  x2,  . . .  xn)         ,    , 
d(xi9x29...xn)      K      }   d(Fl9F2,...  Fn)'      l45; 

d(yi,yz,--  •  yn) 

336.  Again,  if  we  suppose  that  the  equations  connecting 
the  variables  are  transformed,  by  elimination  or  otherwise, 
to  the  following  shape — 

01  (xl9  x29  .  .  .  xm  yi)  =  o, 

<t>2  (#2,  #3,  ...  %n9  2/ij  y*)  =  O, 

^3  (#3,  xi9  ...  xn9  yh  y29  yz)  =  o, 

#»(#»,  yu  ^2,  • .  .  y»)  =  o, 
then  the  Jacobian  determinant 

d{(f>i9  <f>2i  .  .  ■  <j)n) 

d(yi,  2/2,  ...  «/«)' 

as  in  Art.  332,  reduces  to  its  leading  term 

d(f>i  d<j)2  d<fyz  d(pn 

dyx  dy2  dy3  dyn' 


422 

Jacobians. 

In  like 

manner 

d(fa,  fa 
d(xi9  x2 

,  •  •  •  4>n) 

)   •  •  •    <^n) 

reduces  to 

dfa  dfa 
dxi  dx2 

d(pn 

ClXfi 

Accord; 

ingly,  in 

this  case, 

the  Jacobian 
d(f)i  dfa 

dcj> 

d[yi>  2/2,  •  • 

a \X\9  x%)  m . 

.  a?n)      v          dfa  dfa 

Q/Xrfl 

d(pn 

dyx  dyz 

dyn 

(46) 


337.  Case  where  J  =  o. — We  can  now  prove  that  if  the 
Jacobian  vanishes,  the  functions  ylf  yi9 . . .  yn  are  not  indepen- 
dent of  one  another. 

For,  if  J(ylf  y2,  . . .  yn)  =  o,  we  must  have 

dfa  dfa  d(pn 

dxi  dxz        '  dxn         ' 

that  is,  we  have  -p1  =  o  f or  some  value  of  i  between  1  and  n. 

Hence  fa  must  not  contain  xi ;  and  accordingly  the  cor- 
responding equation  is  of  the  form 

fa  (xi+1,  .  .  .  %n,     yly  y2,  . .  .  yi)  =  o. 
Hence  between  this  and  the  remaining  equations, 

fan  =  O,      0i+2  =  0,    ...    (pn  =  O, 

the  variables  asu-u  xt&j  •  •  •  xn  can  be  eliminated  so  as  to  give 
a  final  equation  between  yiy  y2,  . .  .  yn  alone.  This  establishes 
our  theorem. 


Jacobian  of  Implicit  Functions.  423 

338.  In  the  particular  case  where 
yx  =  Fi(xi,  x2,  ...  xn), 
y*  =  Fi{yi,  x*,  ...  xn), 


yn=  Fn{yl,  y%,  ...  h'n-l,  xn), 
we  have 

d{yu  y2,  .  .  .  t/n)  ^cfyi    dy2  dyn 

Ct>  («?ij  X2,  ...  Xn)         CtXi      (X'X<2,  (XtJOn 


(47) 


It  may  be  observed  that  the  theory  of  Jacobians  is  of 
fundamental  importance  in  the  transformation  of  Multiple 
Integrals  (see  Int.  Calc,  Art.  225). 


Examples. 

i.  Find  the  Jacobian  of  yi,  y%,  ...  yn  with  respect  to  r,  9i,  62,  •  ■  ■  0n.i, 
being  given  the  system  of  equations 

yx  =  r  cos  0i,    y2  =  r  sin  0i  cos  62,     yz  =  r  sin  di  sin  62  cos  03,  •  •  • 

yn  =  r  sin  0i  sin  02  ...  sin  dn-\. 
If  we  square  and  add  we  get 

yi2  +  yz2  +  •  •  •  yn2  =  ^2- 
Assuming  this  instead  of  the  last  of  the  given  equations  we  readily  find 

J  =  rn~l  sinM"2  0i  sin"-3  02  •  •  •  sin  0„_2. 
2.  Find  the  Jacobian  of  yi,  t/2,  •  •  •  yn,  being  given 

y\  =  %i  (1  -  #2),    2/2  =  SO1X2  ( 1  -  afo)  .  .  . 

2/n_l  =#i#2  •  ■'.  #w_l(l  -  #»)> 
^n  =  #1  #2  •    .  •  #n. 

Here    2/1  +  2/2  +  •  •  •  yn  =  %i,  and  we  get 

<%i,  y2>  •  •  •  y»)         _,     „  o 

«(#1,  %2>  ■  •  ■  xn) 


424  Jacobians. 

339.  If  ylf  y2,  . .  .  yn,  which  are  given  functions  of  the 
n  variables  Xi,  x2,  . . .  xm  be  conneoted  by  an  independent  re- 
lation 

F{yi,  y2,  . .  •  yn)  =  o,  (48) 

we  may,  in  virtue  of  this  relation,  regard  one  of  the  variables, 
xn  suppose,  as  a  function  of  the  remaining  variables,  and  thus 
consider  yi}  y2,  .  .  .  yn~\  as  functions  of  xXi  o?a,  .  .  .  aw*  In 
this  case  it  can  be  shown  that 

dF_ 
d(Vi>  ^,  ...  yn-i)  _djfad(yl9  y2,  ...  yn) 
d(xi,  a?2,  •  • .  ^Vi)      d_F  e?(a?i,  x2,  ...  #»)" 

For,  if  we  regard  xn  as  a  function  of  o?i,  we  have 
d    .    ,  _ #1      ctyi  dxn       d^        _dy2     dy2  dxn    - 

W$?i  ui^x        ClXift    (IX\         OjX\  dX\        (ZXji    (XX\ 

Also,  from  equation  (48), 

c?i^     e£F  ^  f/i^7      dF  <&„  „ 

h =  O,  +    =  O,  CCC. 

dxi      dxn  dxi  dx2       dxn  dx2 

dF  dF  dF 


«  •  n       j  -v  CIX\  »  QjX2  \  ClXfl—l 

Again,  let    A1=— „    A,  =  3F  . . .  A„_,  =  ^r  ; 


CvXoi  (JjJUiyi,  LvX, 


n  w^n 


then  —  =  -  Ai,     —  =  -  A2,  . . .  -r —  =  -  AM_i. 

W#?l  W#?2  fl'^'Ti— 1 

d  dyx        dyx       d  dyx         dyx 

Hence      —  (2/1)  =  - —  Ai -r- ,    t-  (yi)  =3 —  A3  -7—,  &c. 
rfa?i  <^i        a#w      a#2  dx2         dxn 

&G. 

accordingly,  substituting  in  the  Jacobian 

d(yi,  p29  ♦ .  .  yn-i) 

CC  [Xly      X2)     .     .     .     Xn^lJ 


Jacobian  of  Implicit  Functions. 


425 


it  becomes 

dyx  _      dyx 

OjX\             QjXfi 

dyx      A    ^A 

OjX%              Q/Xi)i 

dyx      .      e&/i 

dy%     x   dy2 

dy%    x  #2 

dy%      .      <%3 

dyn-x      x    %«-i 

dyn-x     ^      wj/^-i 

If  this  determinant  be  bordered  by  introducing  an  addi- 
tional column  as  in  the  following  determinant,  the  other 
terms  of  the  additional  row  being  cyphers,  its  value  is  readily 
seen  to  be 


or 


dyx 

QjX\ 

dyx 
(X1X1 

dyx 

dy* 

dxi 

dy2 
dx2 

dy* 

dyn-x 

dyn-x 

dyn-x 

dxx  ' 

dx%  ' 

(JvXift 

K 

A2,       .  . 

.        1 

dyx 

dxi 

dyx 
dx% 

dyx 

Ci/Xfi 

I 

dy2 

(XX\ 

djh 

dx2 

dy% 

CtXfi 

dF 

• 

. 

(XXi)i 

dy^x 

dyn-x 

dyn-x 

CtX\ 

dx,  ' 

iX/JUryi 

dF 

dF 

dF 

dx* 

(1X2 

CvXm 

426 


Jacobians. 


Again,  we  have 

dF     dF  dy 


dxx 


dF  dy2 
dtfx  dxx      dy%  dxi 


dF     dF  dt/i     dF  dyz 
dx2      dyi  dxz      dy2  dx2 


+ 


+ 


dF  dyn 

dyn  dxi ' 

dF  dyn 

(X'U 'n   UiX<i 


Substituting  these  values  in  the  last  row  of  the  preceding 
the  theorem  is  established,  since  we  readily  find  that  the  de- 
terminant is  reducible  to 


dijx 
dxi 

dyi 

dx% 

dF 
dyn 
dF 

dy* 
dxi 

djh 
dx% 

dy% 

iX/'tA/fl 

• 

• 

• 

dx^ 

dyn 
dx% 

dyn 

Cl/Xji 

(49) 


It  may  be  well  to  guard  the  student  from  the  supposition 
that  this  latter  determinant  is  zero,  as  in  Arts.  333  and  337. 
The  distinction  is,  that  in  the  former  cases  the  equation 
F(yx,  yz,  .  .  .  yn)  =  o,  connecting  the  y  functions,  is  deduced 


by  the  elimination  of  the  variables  xly  os2, 


X-i 


from  the 


equations  of  connexion,  whereas  in  the  case  here  considered 
it  is  an  additional  and  independent  relation. 


Examples.  427 


Examples. 


d2ti 

1.  Being  given  y  =f(u),  and  u  —  #(#),  find  — . 

Am.  /»  j\%)  +f"(u){<t>'(x)\K 

2.  If  y  =  -F(0)  t=f(u),  u  =  <j>(x),  find  the  value  of — 5. 

Am.  F(t)f{u)  <l>"(z)  +  W{x)Y{f'(u)  F\t)  +  (f(u))2F"(l)}. 

3.  Change  the  independent  variable  from  x  to  %  in  the  equation 

.  d2y  o  %       o  '   1  1 

#4  — —  —  2nx6  —  +  a'y  =  o,  where  x  =  -. 

.  <^V    ,     2  (»  +  I)    <&/ 

4.  Transform  (1  -  #2)  —  —  x  —  +  a2y  =  o,  being  given  #  =  sinz. 

.        d2y 
Am.  — —  +  a2y  —  o. 

dz2 

5.  If  Fbe  a  function  of  r,  where  r2  =  #2  +  y2,  prove  that 

d2V     d2V  _d*V      1   ^T 
rf«2        dy2        dr2        r  dr 

6.  If  V  be  a  function  of  r,  where  r2  =  x2  +  y2  +  z2,  prove  that 

d2V     d2V     d2V_^Z      2  ^L 
dx2       dy'*        dz2        dr2        r  dr 

7.  If  #  =  r  sin  0  cos  d>,     v  =  r  sm  0  sin  d>,     2  =  r  cos  0,  prove  that  — ■  =  — , 

dr      dx 

where  in  finding  — ,  6  and  <\>  are  regarded  as  constants ;   while  in  finding 
dr 

— ,  y  and  z  are  regarded  as  constants. 
dx 

8.  If  z  be  a  function  of  two  independent  variables,  x  and  y,  which  are 
connected  with  two  other  variables,  u  and  v,  by  the  equations 

f\ (%,  y,  w,  v)  =  o,        f2(x,  y,u,v)  =  o; 

dz  dz  dz  dz 

show  how  to  express  —  and  —  m  terms  of  -7-  and  — . 


428  Examples. 

9.  Transform  the  equation 


d2y         2x      dy  y 

dx2      1  +  x2  dx      (1  +  x1)2 


=  0 


into  another  in  -which  0  is  the  independent  variable,  supposing  x  =  tan  0. 

d2y 
Am.  MT  +  y  =  o. 

10.  If  z  be  a  function  of  x  and  2/,  and  u  =  px  +  qy  —  z,  prove  that  when 
p  and  q  are  taken  as  independent  variables,  we  have 


du 


dht 


d2u 


dht 


ir  =  %,    T=y,    -J-; 


dp 


dq 


dp1      rt  —  s2'     dp  dq         rt  —  s2 '     dq2       rt  — 


where  p,  q,  r,  s,  t,  denote  the  partial  differential  coefficients  of  z,  as  in  Art.  313. 
ir.  If  the  equation 


dny        .        .  dn~ly 

xn-^  +  AiX*-1- — f  + 

dx11  dx7*'1 


dy 

+  An-lX—  +  An  =  o 

dx 


be  transformed  to  depend  on  0,  where  x  =  ed,  prove  that  the  coefficients  in  the 
transformed  differential  equation  are  all  constants. 

12.  In  orthogonal  transformations,  prove  that 

dV2      dY2      dV2  _  dV2      dV2      dV2 
dx2'  +  dy2   +  dlF  ~  dX2  +  df2  +  ~dZ2  ' 


<b(t)  ib(t) 

13.  Given  x  =  — ,      y  =  j— ^-    prove  that 


d2y 


d2y=( 
dx2       (. 


F(t) 


\F{t)<p'(t)-<p{t)F 
14.  Being  given 


w 


F(t),    F'(t),   F"{t) 
<p(t),     <p'(t),     <p"(t) 

w)>  n*),  v(t) 


find  the  value  of  the  Jacobian 


y\  —  r  sin  d\  sin  02,     y%  =  r  sin  0i  cos  02, 
yz  =  r  cos  0i  sin  03,     y±  =  r  cos  0i  cos  03, 

<*(yi>  y^  yz>  y±) 

Am.  rz  sin  0i  cos  Q\. 


d(r,  0i,  02,  03) 


Examples. 


429 


15.  Find  the  Jacobian  -V^ — - — ;,  being  given 

d(r,  e,  <*>)' 


x  =  r  cos  6  cos  <p,      y  =  r  sin  0  Vi  —  ?w2  sin2<£,      2  =  r  sin<j!>  Vi  —  w2  sin20, 
where    wfi  +  n2  =  1. 


Ans. 


r2  (w2  cos2</>  4-  w2  cos20) 


Vi  —  w2  sin2(£  Vi  —  w2  sin2  0 


16.  Being  given 


X%  X3  X\  Xz  Xi  x% 

v\  =  — ,   y%  = ,    y$  = , 

Xl  xz  xz 


find  the  value  of  the  Jacobian  of  yi,  y%,  yz. 
17.  In  the  Jacobian 

d(yi,  y%,  .  .  .  yn) 

d{X\,  X%,  .  .  .  Xn) 


if  we  make 


prove  that  it  becomes 


yn  = 


Ans.  4. 


u, 

Ml, 

«2,       • 

.  -   Un 

du 

dui 

dui 

dun 

d%\ 

dx\ 

dx\ 

dx\ 

I 
u»+1 

du 
dx% 

du\ 
dx% 

du% 
dx% 

dun 

dx% 

du 

du\ 

ClXii 

du% 

CvXyi 

dun 

dxn 

This  determinant  is  represented  by  the  notation    K(u,  u\,  .  .  .  un). 

18.  If  a  homogeneous  relation  exists  between  u,  u\,  .  .  .  un,  prove  that 

E{u,  «i,  ...  un)  =  o. 

19.  In  the  same  case,  if  yi,  y%,  . .  .  yn  possess  a  common  factor,  so  that 
yi  =  UiU,  &c,  prove  that 

J{yu  yi>  •  •  •  y»)  =  2unJ{ui,  «2,  . . .  «»)  -  w"-1  jt(w,  «i,  .   .  %). 


43o 


Miscellaneous  Examples. 


Miscellaneous  Examples. 

i  .  If  o,  £,  y  be  the  roots  of  the  cubic 

xz  +  px2  +  qx  +  r  =  o, 

dp       dq       dr 

da       da       da 

dp      dq      dr 

df?     ~d&     dp 

dp      dq  ,     dr 
dy'     dy  '     dy 


show  that 


=  (7-jB)(j8-a)(a-7). 


2.  Being  given  the  three  simultaneous  equations 

(pl{%l,  %2,  %3,  #4)  =  O,       $2(21,  %%y  #3,  %l)  =  °>       <M#1,  #2,  Z3,  04)  =  O, 


determine  the  values  of 


dxo      dxz      dx± 
dx\      dx\      dx{ 


3.  If  u  be  a  solution  of  the  differential  equation 

d2V     dW     d2V 
da;4       %3         dz* 

du         du        du        -nii  i  J.-        £  •;- 

T>rove  that     x V  V  ^r  +  z  —     -will  also  be  a  solution  of  it. 

r  dx        dy        dz 

4.  If  x  and  y  be  not  independent,  prove  that  the  equation  — — -  = 
does  not  hold,  in  general.  axaV      a^ax 

5.  Prove  that  the  points  of  intersection  of  a  curve  of  the  fourth  degree  with 
its  asymptotes  lie  on  a  conic  ;  and  in  general  for  a  curve  of  the  degree  n  they 
lie  on  a  curve  of  the  degree  n  —  2. 

6.  Prove  that  every  curve  of  the  third  degree  is  capable  of  being  projected 
into  a  central  curve.     (Chaales.) 

For  if  the  harmonic  polar  of  a  point  of  inflexion  be  projected  to  infinity,  the 
point  of  inflexion  will  be  projected  into  a  centre  of  the  projected  curve  {see  p.  282). 

7.  Two  ellipses  having  the  same  foci  are  described  infinitely  near  one 
another;  how  does  the  interval  between  them  vary? 

(a).  How  will  the  interval  vary  if  the  ellipses  be  concentric,  similar,  and 
similarly  placed  ? 

8.  Eliminate  the  arbitrary  functions  from  the  equation  z  =  ${x)  .  $(y). 

9.  Show  that  in  order  to  eliminate  n  arbitrary  functions^  from  an  equation 
containing  two  independent  variables,  it  is,  in  general,  requisite  to  proceed  to 
differentials  of  the  order  2n  -  1.  How  many  resulting  equations  would  be  ob- 
tained in  this  case  ? 


Miscellaneous  Examples.  431 

10.  In  the  Lemniscate  r%  =  a%  cos  20,  show  that  the  angle  between  the  tan- 
gent  and  radius  vector  is  -  +  20. 

2 

ii.  In  transforming  from  rectangular  to  polar  coordinates,  prove  that 


12.  Prove  that  the  ellipses 

«V  +  b*x2  =  aW  (i),     a?x2  sec4(p  +  %2  cosec4<p  =  «M  (2), 

are  so  related  that  the  envelope  of  (2)  for  different  values  of  (p  is  the  e volute  of 
(1);  and  the  point  of  contact  of  (2)  with  its  envelope  is  the  centre  of  curvature 
at  the  point  of  (1)  whose  excentric  angle  is  <p. 

13.  Being  given  the  equations 

bx  =  XfjL,     by  =  ^/(A2-*2)^2-^2), 
prove  that 


dx2 


14.  If  1  -  y  —  aym  —  o,  develop  yr  in  terms  of  a  by  Lagrange's  Theorem. 

15.  Being  given  x  =  r  cos  0,  y  =  r  sin  0,  transform 


{-(ST 


d*y_ 
dx2 


into  a  function  of  r  and  9,  where  0  is  taken  as  the  independent  variable. 


Ans. 


d'lr        (dr\ 

d¥*2  \d0) 


16.  Apply  the  method  of  infinitesimals  to  find  a  point  such  that  the  sum  of 
its  distances  from  three  given  points  shall  be  a  minimum. 

Let  pi,  pz,  ps  denote  the  three  distances,  and  we  have  dpi  +  dpi  +  dpz  =  o  : 
suppose  dpi  =  o,  then  d(po  +  p3)  =  o,  and  it  is  easily  seen  that  pi  bisects  the 
angle  between  p%  and  pz,  and  similarly  for  the  others  ;  therefore,  &c. 

j\  'f.  17.  Eliminate  the   circular  and  exponential  function  from  the  equation 
y  =  esin"  *. 

18.  One  leg  of  a  right  angle  passes  through  a  fixed  point,  whilst  its  vertex 
slides  along  a  given  curve ;  show  that  the  problem  of  finding  the  envelope  of 
the  other  leg  of  the  right  angle  may  be  reduced  to  the  investigation  of  a  locus. 


432  Miscellaneous  Exam/pies. 

19.  If  two  pairs  of  conjugates,  in  a  system  of  lines  in  involution,  be  given 
by  the  equations 

u  =  ax2  +  2bxy  +  cy2  =  0,     u  =  dx2  +  zb'xy  +  cy2  =  o, 
show  that  the  double  lines  are  given  by  the  equation 

du  du       du  du'  ICi  .        ,    „    .       .   ,  . 

— —  =  o.     (Salmon  s  Comes,  Art.  342.) 

dx  dy       dy  dx 

20.  It  U\  =  — ,      u%  =  — ,      Un-i  = , 

Xn  %n  %n 

where  xi,  x%,  ...  xn  are  connected  by  the  relation 

%i2  +  #22  +  %z2  +  •  •  •  .+  xn2  =  1, 
prove  that  the  Jacobian 

d(U\,  Uo,    ...    Mn-l)  _        I 
d(X\,  X2,    ...    Xn-l)        Xnn+l' 

21.  If  the  variables  yi,y2,  •  •  .  yn  are  related  to  xi,  #2,  •  •  .  xn  by  the 
equations 

yx  =a\X\  +  a%xz  +  .  .  .  +  anxn, 

y2  =  bixi  +  b%x2  +  .  .  .  +  £„#„, 


2/»  =  h  C0\   +  ?2  #2  +  •   •  •    +  In  Xni 


and  we  have  also 


X12  +  #2*  +  •   •   •  +  #n2  =  I, 

2/12  +  y%2  +  •  •  •  +  yn  =  I, 
prove  that  the  Jacobian 

<%i»  2/2,  ■  .  .  yn-i)  _  y^ 
«?(«i,  #2,  •  •  •  #«-i)      #»* 

22.  Prove  that  the  equation 

r«/2  —  2sxy  +  tx2  =  px  +  qy  —  z 

d2z 
may  be  reduced  to  the  form  —  -f  z  =  o  by  putting  x  =  u  cos  v,     y  =  u  sin  v. 

23.  Investigate  the  nature  of  the  singular  point  which  occurs  at  the  origin 
of  coordinates  in  the  curve 

#4  —  2ax2y  —  axy2  +  a2y2  =  o. 

1  1 

24.  Investigate  the  form  of  the  curve  represented  by  the  equation  y  =  e  x 


Miscellaneous  Examples.  433 

'25.  How  -would  you  ascertain  whether  a  proposed  expression,  V,  involving 
x,  y,  and  z,  is  a  function  of  two  linear  functions  of  these  same  variables  ? 

Am.  The  given  function  must  be  homogeneous  ;  and  the  equations 

dV  _       dV  _       dV  _ 
dx  dy         '  dz 

must  be  capable  of  being  satisfied  by  the  same  values  of  x,  y,  z :  i.  e.  the  result 
of  the  elimination  of  x,  y,  and  z  between  these  equations  must  vanish  identi- 
cally. 

26.  If  y  =  <p(x2),  prove  that 

— -  =  {2xn)  <£(") (x2)  +n(n-i)  {2x)n~2  ft*'1) (xz) 
(too 

n(n- i)(n  —  2)(n  —  3)     s    a    ,   .»  /  ox    „ 

+  - - —  2z)«"4d>(n~2)(>2  >  &c. 

1.2  /       r         \     /' 

27.  If  x  +  iy  =  (a  +  i/3)w,  where  i  =  V-  I,  prove  that 


dx2+dy2       0da2  +  d&2 

. =  ni . 

x*  +  y2  a2  +  j8a 


1  j-,    >  dd>         /i  —  e2  sin2 

^zrrr  prove  that  —  +  J a-^Ti 

^/j      c3  #      >/ 1  -  c2  srn2»J/ 


1  „     .  »<p  /i  —  C    Sill"© 

28.  If  tan  <£  tan  \p  =  .    prove  that  —  +  J =  o. 


29.  If  a;  =  — ,  prove  that 
Jcy 


transforms  into 


d  \  du        du  d2u 

dx  dx2 ' 


V(i  -  x2)  (1  -  h2x2)  V(i  -  y2)  (1  -  h2y2) 

d  I  d\ 

30.  Prove  that  —  (xu)  =  li+x—ju. 

31.  Hence  prove  that 

(   JL\  (x~-i\  u  =  x2  — 
\   dx)  \    dx       )  dx2' 

For  (4)(41=(*  +  <4)|  = 

\    dx)  \    dx)     \  dx)  dx 

(d  \  I    du\        nd2u 

32.  Prove  that 

\   dx)  \   dx        )  \   dx       ) 

By  the  preceding  example  we  have 

\    d%~    }  \   dx       J  \    dx)  \    dx       j 


dxs ' 


xzd  u 


d2u 
dz?' 
2  F 


434 

tut 
therefore 


Miscellaneous  Examples. 

d3u  d2u 


d  I     d2u\        „  dAu  dlu 

dx  \     dx2  J  dx3  dx2 

I     d  \    „< 

£ 2)  X2- 

\    dx         I 


d2u 
dx2 


d3u 
dx3' 


33.  Prove,  in  general,  that 

[x—\[x—-i\\x—-2]...  [x  —  -n  +  i]u  =  xn 
\    dx)  \    dx        J  \    dx       }  \    dx  / 


dnu 
dxn' 


This  can  be  easily  arrived  at  from  the  preceding  by  the  method  of  mathematical 
induction  ;  that  is,  assuming  that  the  theorem  holds  for  any  positive  integer  n, 
prove  that  it  holds  for  the  next  higher  integer  (n  +  1),  &c. 


34.  Find  -  + 


de* 


in  terms  of  r  when  r2  =  a2  cos  2d. 


Am. 


3^ 

y5 


35.  If  u  —  (x2  +  y2  +  22)*,  prove  that 

d*u      dhi      d^u  d*u  d*u  d^u 

dx^+  dyT  +  d?+2  dx2dy2  +  2  dy2dz2  +  2  dz2dx2  =  °' 


36    Iiz  = 


xl  +  y 


dnz      ,       .     1  .  2  .  3 

—  =  (-i)» * 

dx"      v 


,  and  <p  =  tan-1  ( - ) ,  prove  that 

n .  cos(w  +  1)  <p  .  co&n+1<f> 


CM+1 


^2n2  I.2.3.  •  •  2n  ■  cos  (2W  +  *)  0  •  COS2""1^) 


%2»»+1 


1.2.3...  {2n  +  J)  sm  (2^  +  2)  0  •  COS2w+20 

=  \       */  ,v2«.+2 


37.  If  u  be  a  homogeneous  function  of  the  nth  degree  in  x,  y,  z,  and  wi,  #2,  #3, 
denote  its  differential  coefficients  with  regard  to  x,  y,  z,  respectively,  while 
#n,  2*12,  &c.,  in  like  manner  denote  its  second  differential  coefficients  ;  prove  that 


#11,  #12,  #13,  #1 

#21,  #22,  #23,  #2 

#31,  #32,  #33,  #3 

#1,  #2,  #3,  O 


n  —  1 


Mil,  #12,  #13 
#21,  #22,  #23 
#31,    \  #32,       #33 


Miscellaneous  Examples. 


435 


38.  If  u  be  a  homogeneous  function  of  the  nth  degree  in  x,  y,  z,  w,  show 
that  for  all  values  of  the  variables  which  satisfy  the  equation  u  =  o  we  have 


w~ 


(n  -  if 


Wil,  U12,  U13,  ttu 

W215  W22,  2<23,  ^24 

%lj  W32,  ^33,  W34 

%U,  W42,  W43,  W44 


Wli,  W12,  Wl3,  Wi 

#21,  #22,  #23,  #2 

#31,  #32,  #33,  #3 

#1,  #2,  «3,  O 

39.  Show  that  the  equation 

d   (  9N  dP)  1      ^p 

is  satisfied  if  P  is  any  of  the  quantities 

-  -  /jfi,  (1  -  /*2)  cos 20,  (1  -  fj)  sin  20,  fi^/i  -  fjfi  cos 0,  ^  Vi  -  fx2  sin 0, 

or  any  linear  function  of  them. 

40.  If  x  4-  A  be  substituted  for  x  in  the  quantic 

,     n(n  —  1) 
a0xn  +  naixn~1  +  — - -  a%xn~2  +  &c.  +  an, 

and  if  a'o,  «'i,  ....«'*•••••  denote  the  corresponding  coefficients  in  the  new 
quantic ;  prove  that 

da'r        , 
—-  =  ra  r-i. 

It  is  easily  seen  that  in  this  case  we  have 

r(r  —  1) 


a'r  =  ar+  rar-\  X  + 


1  .  2 


«r-2A2  +  &C. 


+  a0\r; 


&c. 


41.  If  4>  be  any  function  of  the  differences  of  the  roots  of  the  quantic  in  the 
preceding  example,  prove  that 


Id  d 

\     aai  da% 


d_ 
daz 


+ 


+  na% 


d  \ 

dan) 


This  result  follows  immediately,  since  any  function  of  the  differences  of  the 
roots  remains  unaltered  when  x  +  A  is  substituted  for  x,   and  accordingly 

d<b  ,  . 

—  =  o  m  this  case. 
d\ 

42.  Being  given 

u  =  xy  +  *y  1  —  x% -  y2  +  x2y2,    <v  =  X'\/i-y'l-\-y^/i  -  x2, 


2F2 


436  Miscellaneous  Examples. 

prove  that 

du  dv      dv  du 

=0, 

dx  dy      dx  dy 

and  explain  the  meaning  of  the  result. 
43.  Find  the  ininimuin  value  of 

sin  A  sin-B-  sin  C 

+ : — -  +  —. — ^— : — -,  where  A  +  B  +  C  =  1800 


sin  B  sin  G     sin  C  sin  A      sin  A  sin  B 

44.  Prove  that 

■where  <p  (x)  is  a  rational  function  of  x. 

45.  Show  that  the  reciprocal  polar  to  the  evolute  of  the  ellipse 

x2     y2 
— \-  —  =  1 
a2^  b2      *> 

with  respect  to  the  circle  described  on  the  line  joining  the  foci  as  diameter,  has 
for  its  equation 

a2      b2 

a>      y* 

46.  If  the  second  term  be  removed  from  the  quantic 

(«o,  ai,  fl2,  . . .  «n)  (#,  y)n 

bv  the  substitution  of  a; y,  instead  of  a;,  and  if  the  new  quantic  be  denoted 

by  (Ao,  o,  A%,  A3,  .  .  .  An)  [x,  y)  ;  show  that  the  successive  coefficients 
A2,  A3,  .  .  .  An  are  obtained  by  the  substitution  of  a\  for  x  and  —  oq  for  y  in 
the  series  of  quantics 

(a0,  ai,  az)  (x,  y),     {a0)  «i,  a2,  «s)  (#,  y),  .  .  .  (#0,  «i,  .  •  •  «n)  («,  y)- 
47.  Distinguish  the  maxima  and  minima  values  of 

1  4-  2x  tan-1  a; 

1+a;2       ' 

a'a;2  +  2&'a;  +  c  ,_    . 

1  ^y  _  (ac  -  b2)  y2  +  (ae'  +  a'c  -  2bb')y  +  a'c  -  b'2 

2  «#  (ab') x2  —  (ca') x  +  {be) 


Miscellaneous  Examples. 


437 


49.  IflX  +  mY+nZ,  l'X  +  m'Y+  n'Z,  l"X  +  m"Y  +  n"Z,  be  substituted 
for  x,  y,  z,  in  the  quadratic  expression  ax2  +  by2  +  cz2  +  2dyz  +  2ezx  +  2fxy ; 
and  if  a',  b\  c',  d\  e',  /',  be  the  respective  coefficients  in  the  new  expression ; 
prove  that 


e\     d\     c', 


=  o  whenever 


a, 

f, 

e 

f, 

h 

d 

e, 

d, 

c 

=  0* 


50.  If  the  transformation  be  orthogonal,  i.  e.  if 

&  +  yi  +  S2  =  X2  +  Y2  +  £2, 

prove  that  the  preceding  determinants  are  equal  to  one  another. 

5 1 .  Prove  that  the  maximum  and  minimum  values  of  the  expression 

ax^  +  $bxz  —  6cx2  +  \dx  +  e 
are  the  roots  of  the  cubic 

a3s3  -  3  {a2!  -  3#2)  z2  +  3  («12  -  I8J57)  2  -  A  =  o, 

where  II  =  ac  —  b2,     I  *=  ae  —  ^bd  +  3c2, 


/  = 


a,  b,     c 

b,  c}     d 
e,     d,     e 


,  and  A  =  I3  -  27J"2. 


By  Art.  138  it  is  evident  that  the  equation  in  z  is  obtained  by  substituting 
e  —  z  instead  of  e  in  the  discriminant  of  the  biquadratic  ;  accordingly  we  have 
for  the  resulting  equation 

(J-  azf  =27(7-  zEfj 
since  the  discriminant  of  the  biquadratic  is 

J3  _  27/2  _  o. 

In  general,  the  equation  in  z  whose  roots  are  the  n  —  1  maximum  and  mini- 
mum values  of  a  given  function  of  n  dimensions  in  x,  can  be  got  from  tbe  dis- 
criminant of  the  function,  by  substituting  in  it,  instead  of  the  absolute  term, 
the  absolute  term  minus  z. 

It  is  evident  that  the  discriminant  of  the  function  in  x  is,  in  all  cases,  the 
absolute  term  in  the  equation  in  z. 

52.  If  A  be  the  product  of  the  squares  of  the  differences  of  the  roots  of 


#3  —  px2  -f  qx  —  r  =  o, 


43  8  Miscellaneous  Examples. 

find  an  expression  in  terms  of  the  roots  for  — ,  by  solving  from  three  equations 

of  the  form 

dA      dA  dp      dA  dq      dA  dr 
da       dp  da      dq  da      dr  da' 

Ans.  2  (£  +  7  -  2a)  {y  +  a  -  20)  (a  +  0  -  27). 

53.  If  X  +  Y*y -  1  be  a  function  of  x  +  y  */  —  1,  prove  that  X  and  Y 
satisfy  the  equations 

d*X     d*X              ,  d2Y     d*Y 
H 5-  =  o,  and 1 =  o. 

dx2       dy-  dx%        dy1 

54.  If  the  three  sides  of  a  triangle  are  a,  a  +  a,  a  +  /3,  where  a  and  £  are 
infinitesimals,  find  the  three  angles,  expressed  in  circular  measure. 

.        7r      a  +  fi      it      2a  — /J      w      2fi  —  a 
Ans. z,     -  H Z.J     -  + -. 

3      ciy/ ?>      3      a\/z      ^      ay^s 

55.  If  2/  =  #  +  ax3,  where  a  is  an  infinitesimal,  find  the  order  of  the  error  in 
taking  x  —  y  —  ay3. 

56.  The  sides  a,  b,  c,  of  a  right-angled  triangle  become  a  -t-  a,  b  +  /3,  c  +  7, 
where  a,  )8,  7  are  infinitesimals  ;  find  the  change  in  the  right  angle. 

.        cy  —  aa  —  bfi 

Ans. -. 

ab 

57.  If  a  curve  be  given  by  the  equations 

2x  =  \/t2  +  2t  +  \/p  -  2*, 
2y  =  <y  t*  +  2t  -\/t~  -  2t, 

find  the  radius  of  curvature  in  terms  of  t. 

58.  In  the  curve  whose  equation  is  y  =  e-*2,  determine  all  the  cases  where 
the  tangent  is  parallel  to  the  axis  of  x. 

If  0  be  the  greatest  angle  which  any  of  its  tangents  makes  with  the  axis  of  x, 


prove  that  tan  0 


-£ 


59.  In  a  curve  traced  on  a  sphere,  prove  the  following  formula  for  the  radius 
of  curvature  at  any  point  : 

sinrdr 
tan  p  = . 

COS  pdp 

60.  Apply  this  form  to  show  that  in  a  spherical  ellipse  sin  p  sinp'  =  const., 
where  p  and  p  are  the  perpendiculars  from  the  foci  on  any  great  circle  touching 
the  ellipse. 


Miscellaneous  Examples.  439 

61.  Prove  the  following  relation  between  (p,  p),  the  radii  of  curvature  at 
corresponding  points  of  two  reciprocal  polar  curves  : 

99         COS3^/' 

where  \p  is  the  angle  between  the  radius  vector  and  normal. 

62.  If  AB,  BC,  CD,  ...  be  the  sides  of  an  equilateral  polygon  inscribed  in 
any  curve,  and  if  AB  be  produced  to  meet  B  G  in  F ;  prove  that,  when  the  sides 

P2 
of  the  polygon  are  diminished  indefinitely,  BP  =3^-7,  where  p  and  p'  are  the 

.  P 
radii  of  curvature  at  B  and  at  the  corresponding  point  of  the  evolute. 

63.  if     u=  */(*  -  x) (r  +  y  +  y2)  +  \A*  - y) (r  +  x  +  x2\ 

x-y 


and  V  =  ( - —  )    +  x  +  y, 

\  x-y  1 


find  the  value  of 


dUdV     dVdTT 
dx  dy       dx  dy 


64.  If  V—  xn  +  — ,     and    s  =  x  +  -, 

iC»  x 


prove  that 


._       .rf2F        dV      3T_ 


65.  Determine  b  and  &  so  that  the  curve 

(x2  +  2/2)  (x  cos  a  +  y  sin  a  -  0)  =  W  {x  cos  3  +  y  sin  &  —  b) 

may  have  a  cusp ;  a,  j8,  and  a  being  given  and  the  coordinates  being  rectan- 
gular. 

Prove  that  in  this  case  the  cuspidal  tangent  makes  equal  angles  with  the 
asymptote  and  with  the  line  drawn  from  the  cusp  to  the  origin. 

66.  Find  the  coordinates  of  the  two  real  finite  points  of  inflexion  on  the 
curve  y1  =  (x  —  2)2  (x  —  5),  and  show  that  they  subtend  a  right  angle  at  the 
double  point. 

67.  If  x,  y,  z,  be  given  in  terms  of  three  new  variables  u,  v,  w,  by  the  fol- 
lowing equations  :  x  =  Pu,  y  =  (P  —  b)v,  z  =  (P  —  c)w,  where 

1  +  bv2  +  ew2 

ifi  +  V2  +  w2  ' 


44°  Miscellaneous  Examples. 

it  is  required  to  prove  that  dx2  +  dy2  +  dz2  =  Z2du2  +  M2dv2  +  N2dw2,  and  to 
determine  the  actual  values  of  Z,  M,  N. 

68.  If  x  +  y  =  X,     y  =  XT,  prove  that 

d2u  d2u       du      ^d2u       „   d2u        du 

<fe3      y  dkdy      dx~      dX2         dXdY     dX 

69.  Being  given  x  =  uz  -  3m2,  y  =  xu2v  -  #3,  find  what  —       ^ —  becomes 
in  terms  of  u,  v,  du,  dv.  xdx  +  VdV 

.       udv  —  vdu 

Ans.  — -. 

udu  +  v  dv 

a 

70.  If  the  polar  equation  of  a  curve  he  r  =  a  sec2-,  find  an  expression  for 
its  radius  of  curvature  at  any  point. 

doc 

71.  Show  that  the  differential  —  is  transformed  into 

yV  -  yx2  +  3 
Idy 


V(i  +  if  tan2 A)  (1  +  y2cot2\)' 


4  /—  l  —  y 
by  assuming  a;  =*/  3 ,  and  find  the  value  of  A. 


Ans.  A  =  7° 30'. 


72.  If  y4  +  xy  =  1,  prove  that 

„  d2y  dyd        dy2 

y2  — -  +  xx  —  +  y  —  =  o. 
y  dx2  ^  6    dx*     y  dx2 

73.  The  pair  of  curves  represented  by  the  equation 

r2  —  2rF(oj)  +  c2  =  o 

may  be  regarded  as  the  envelope  of  a  series  of  circles  whose  centres  lie  on  a 
certain  curve,  and  which  cut  orthogonally  the  circle  whose  radius  is  e,  and 
whose  centre  is  the  origin  (Mannheim,  Journal  de  Math.,  1862). 

74.  A  chord  PQ  cuts  off  a  constant  area  from  a  given  oval  curve ;  show  that 
the  radius  of  curvature  of  its  envelope  will  be  ^PQ  (cot  9  +  cot  <£>),  0  and  <p  being 
the  angles  at  which  PQ  cuts  the  curve. 

75.  In  the  polar  equations  of  two  curves, 

F(r,  «)  =  o,    f{r,  »)=o, 

if  Rtn  be  substituted  for  r,  and  nO.  for  &>,  prove  that  the  curves  represented  by 
the  transformed  equations  intersect  at  the  same  angle  as  the  original  curves. 

(Mr.  W.  Roberts,  ZiouvilWs  Journal,  Tome  13,  p.  209). 


Miscellaneous  Examples.  441 

This  result  follows  immediately  from  the  property  that  —  is  unaltered  by 
the  transformation  in  question. 

76.  A  system  of  concentric  and  similarly  situated  equilateral  hyperbolas  is 
cut  by  another  such  system  having  the  same  centre,  under  a  constant  angle, 
which  is  double  that  under  which  the  axes  of  the  two  systems  intersect. 

Ibid.,  p.  210. 

77.  In  a  triangle  formed  by  three  arcs  of  equilateral  hyperbolas,  having  the 
same  centre  (or  by  parabolas  having  the  same  focus),  the  sum  of  the  angles  is 
equal  to  two  right  angles.  Ibid.,  p.  210. 

78.  Being  given  two  hyperbolic  tangents  to  a  conic,  the  arc  of  any  third 
hyperbolic  tangent,  which  is  intercepted  by  the  two  first,  subtends  a  constant 
angle  at  the  focus.  Ibid.,  p.  212. 

An  equilateral  hyperbola  which  touches  a  conic,  and  is  concentric  with  it,  is 
called  a  hyperbolic  tangent  to  the  conic. 

79.  A  system  of  confocal  cassinoids  is  cut  orthogonally  by  a  system  of  equi- 
lateral hyperbolas  passing  through  the  foci  and  concentric  with  the  cassinoids. 

Ibid.,  p.  214. 

The  student  will  find  a  number  of  other  remarkable  theorems^  deduced  by 
the  same  general  method,  in  Mr.  Eoberts'  Memoir.  This  method  is  an  exten- 
sion of  the  method  of  inversion. 

80.  If  Pn  be  the  coefficient  of  xn  in  the  expansion  of  (1  -  2ax  +  x2)-',  prove 
the  two  following  equations  : 

dPn 
(a2  —  1)  — —  =  naPn  —  nPn-\, 
da 

nPn  =  {2n  -  1)  aPn.i  -  (n  -  1)  PM_2. 

81.  If  at  each  point  on  a  curve  a  right  line  be  drawn  making  a  constant 
angle  with  the  radius  vector  drawn  to  a  fixed  point,  prove  that  the  envelope  of 
the  line  so  drawn  is  a  carve  which  is  similar  to  the  negative  pedal  of  the  given 
curve,  taken  with  respect  to  the  fixed  point  as  pole. 

82.  If  2  U  =  ax1  +  2bxy  +  eg2,     2  V  =  a'x2  +  2b' xy  +  e'y2, 

=  AU2  +  2BUV+  CV2,  find  A,  B,  C. 


and 


dU  dU 

dx'  dy 

dV  dV 

dx '  dy 


83.  Prove  that  the  values  of  the  diameters  of  curvature  of  the  curve  y2  =f(x) 
where  it  meets  the  axis  of  x  are  /'(«)»  /'(#)>  ....  if  a,  /8,  ...  be  the  roots  of 
/(*)=o. 

Hence  find  the  radii  of  curvature  of  y2  =  (x2  -  m2)  (x  —  a)  at  such  points. 

84.  A  constant  length  PQ  is  measured  along  the  tangent  at  any  point  P  on 
a  curve  ;  give,  by  aid  of  Art.  290,  a  geometrical  construction  for  the  centre  of 
curvature  of  the  locus  of  the  point  Q. 


44 2  Miscellaneous  Examples. 

85.  In  same  case,  if  PQ[  be  measured  equal  to  PQ,  in  the  opposite  direction 
along  the  tangent,  prove  that  the  point  P,  and  the  centres  of  curvature  of  the 
loci  of  Q,  and  Q',  lie  in  directum. 

86.  A  framework  is  formed  by  four  rods  jointed  together  at  their  extremities  ; 
prove  that  the  distance  between  the  middle  points  of  either  pair  of  opposite  sides 
is  a  maximum  or  a  minimum  when  the  other  rods  are  parallel,'  being  a  maximum 
when  the  rods  are  uncrossed,  and  a  minimum  when  they  cross. 

87.  At  each  point  of  a  closed  curve  are  formed  the  rectangular  hyperbola, 
and  the  parabola,  of  closest  contact ;  show  that  the  arc  of  the  curve  described  by 
the  centre  of  the  hyperbola  will  exceed  the  arc  of  the  oval  by  twice  the  arc  of 
the  curve  described  by  the  focus  of  the  parabola ;  provided  that  no  parabola  has 
five-pointic  contact  with  the  curve.     (Gamb.  Math.  Trip.  1875.) 

88.  A  curve  rolls  on  a  straight  line,  determine  the  nature  of  the  motion  of 
one  of  its  involutes.     (Prof.  Crofton.) 

89.  Prove  the  following  properties  of  the  three-cusped  hypocycloid :  — 

(1).  The  segment  intercepted  by  any  two  of  the  three  branches  on  any 
tangent  to  the  third  is  of  constant  length.  (2).  The  locus  of  the  middle  point 
of  the  segment  is  a  circle.  (3).  The  tangents  to  these  branches  at  its  extremities 
intersect  at  right  angles  on  the  inscribed  circle.  (4).  The  normals  corresponding 
to  the  three  tangents  intersect  in  a  common  point,  which  lies  on  the  circum- 
scribed circle. 

Definition. — The  right  line  joining  the  feet  of  the  perpendiculars  drawn  to 
the  sides  of  a  triangle  from  any  point  on  its  circumscribed  circle  is  called  the 
pedal  line  of  the  triangle  relative  to  the  point. 

90.  Prove  that  the  envelope  of  the  pedal  line  of  a  triangle  is  a  three-cusped 
hypocycloid,  having  its  centre  at  the  centre  of  the  nine-point  circle  of  the 
triangle.  (Steiner,  JJeber  eine  besondere  curve  dritter  klasse,  und  vierten  grades, 
Crelle,  1857.) 

This  is  called  Steiner'' s  Envelope,  and  the  theorem  can  be  demonstrated, 
geometrically,  as  follows  : — 

Let  P  be  any  point  on  the  circumscribed  circle  of  a  triangle  ABO,  of  which  D 
is  the  intersection  of  the  perpendiculars  ;  then  it  can  be  shown  without  difficulty, 
that  the  pedal  line  corresponding  to  P  passes  through  the  middle  point  of  DP. 
Let  Q  denote  this  middle  point,  then  Q  lies  on  the  nine-point  circle  of  the 
triangle  ABC.  If  0  be  the  centre  of  the  nine-point  circle,  it  is  easily  seen  that, 
as  Q  moves  round  the  circle,  the  angular  motion  of  the  pedal  line  is  half  that  of 
OQ,  and  takes  place  in  the  opposite  direction.  Let  R  be  the  other  point  in 
which  the  pedal  line  cuts  the  nine-point  circle,  and,  by  drawing  a  consecutive 
position  of  the  moving  line,  it  can  be  seen  immediately  that  the  corresponding 
point  T  on  the  envelope  is  obtained  by  taking  QT  =  QR.  Hence  it  can  be 
readily  shown  that  the  locus  of  T  is  a  three-cusped  hypocycloid. 

This  can  also  be  easily  proved  otherwise  by  the  method  of  Art.  295  (a). 

91.  The  envelope  of  the  tangent  at  the  vertex  of  a  parabola  which  touches 
three  given  lines  is  a  three-cusped  hypocycloid. 

92.  The  envelope  of  the  parabola  is  the  same  hypocycloid. 


On  the  Failure  of  Taylor's  Theorem.  443 

For  fuller  information  on  Steiner's  Envelope,  and  the  general  properties  of 
the  three-cusped  hypocycloid,  the  student  is  referred,  amongst  other  memoirs,  to 
Cremona,  Crelle,  1865.  Townsend,  Educ.  Times.  Reprint.  1866.  Ferrers, 
Quar.  Jour,  of  Math.,  1866.  Serret,  Now.  Ann.,  1870.  Painvin,  ibid.,  1870. 
Cahen,  ibid.,  1875. 


On  the  Failure   of   Taylor's   Theorem. 

As  no  mention  has  been  made  in  Chapter  III.  of  the  cases  when  Taylor's 
Series  becomes  inapplicable,  or  what  is  usually  called  the  failure  of  Taylor's 
Theorem,  the  following  extract  from  M.  Navier's  Legons  d' Analyse  is  intro- 
duced for  the  purpose  of  elucidating  this  case  : — 

On  the  Case  when*  for  certain  particular  Values  of  the 
Variable,  Taylor's  iSeries  does  not  give  the  Uevelopinent  of 
the  Function. — The  existence  of  Taylor's  Series  supposes  that  the  function 
f(x)  and  its  differential  coefficients  f(x),  f"(x),  &c,  do  not  become  infinite  for 
the  value  of  x  from  which  the  increment  h  is  counted.  If  the  contrary  takes 
place,  the  series  will  be  inapplicable. 

Fix) 

Suppose,  for  example,  that/(#)  is  of  the  form  - — K—^-,  m  being  any  positive 

number,  and  Fix)  a  function  of  x  which  does  not  become  either  zero  or  infinite 
when  x  =  a. 

Fix  +  h) 

If,  conformably  to  our  rules,  - — y— V-  be  developed  in  a  series  of  posi- 

'  J  '  ix  +  h  -  a)m 

tive  powers  of  h,  all  the  terms  would  become  infinite  when  we  make  x  =  a.    At 

F(a  +  h)      _   t 
the  same  time  the  function  has  then  a  determinate  value,  viz.  :  — — — .     -But 

hm 

as  the  development  of  this  value  according  to  powers  of  h  must  necessarily  con- 
tain negative  powers  of  h,  it  cannot  be  given  by  Taylor's  Series. 

Taylor's  Series  naturally  gives  indeterminate  results  when,  the  proposed 
function  fix)  containing  radicals,  the  particular  value  attributed  to  x  causes 
these  radicals  to  disappear  in  the  function  and  in  its  differential  coefficients. 
In   order  to  understand  the  reason,   we  remark  that  a  radical  of  the  form 

p 
ix  -  df,  p  and  q  denoting  whole  numbers,  which  forms  part  of  a  function /_(#), 
gives  to  this  function  q  different  values,  real  or  imaginary.  As  this  same  radical 
is  reproduced  in  the  differential  coefficients  of  the  function,  these  coefficients  also 
present  a  number,  q,  of  values.  But,  if  the  particular  value  a  be  attributed  to <%, 
the  radical  will  disappear  from  all  the  terms  of  the  series,  while  it  remains 

p 
always  in  the  function,  where  it  becomes  hq.  Therefore  the  series  no  longer  re- 
presents the  function,  because  the  latter  has  many  values,  while  the  series  can 
have  but  one.  The  analysis  solves  this  contradiction  by  giving  infinite  values 
to  the  terms  of  the  series,  which  consequently  does  not  any  longer  represent  a 
determined  result. 

The  development  of  f(x)  ought,  in  the  case  with  which  we  are  occupied,  to 
p 
contain  terms  of  the  form  hq.     We  should  obtain  the  development  by  making 
x  =  a  +  h  in  the  proposed  function. 


444  On  the  Failure  of  Taylor's  Theorem. 

Fractional  powers  of  h  would  appear  in  the  latter  development :  for  example, 
suppose 

f(x)  =  2ax  —  x2  —  a  y  x2  —  a2  ; 

this  gives 

(too 

/»  =  2  («-*)  + 


/»=-2  + 


*/x2-a2      {x2-a2Y 

On  making  x  —  a,  we  have  fix)  =  a2,  and  all  the  differential  coefficients 
"become  infinite.  This  circumstance  indicates  that  the  development  of  f(x  +  h) 
ought  to  contain  fractional  powers  of  h  when  x-a'.'vo.  fact  the  function  be- 
comes then 

f(a  +  h)  =a2-  h2-{-  a^2ah-\-  h2, 

of  which  the  development  according  to  powers  of  h  would  contain  hi,  h*,  ifi,  &c. 

It  should  be  remarked  that  a  radical  contained  in  the  function  f(x)  may- 
disappear  in  two  different  ways  when  a  particular  value  is  attributed  to  the 
variable  x,  that  is,  i°,  when  the  quantity  contained  under  the  radical  vanishes: 
20,  when  a  factor  with  which  the  radical  may  be  affected  vanishes. 

In  the  former  case  the  development  according  to  Taylor's  Theorem  can  never 
agree  with  the  function  f(x  +  h)  for  the  particular  value  of  x  in  question,  for 
the  reason  already  indicated. 

Eut  it  is  not  the  same  in  the  latter  case,  because  the  factor  with  which  the 
radical  is  affected,  and  which  becomes  zero  in  the  function,  may  cease  to  affect 
the  radical  in  the  differential  coefficients  of  higher  orders ;  in  fact  it  may  not 
disappear  at  all,  and  the  series  may  in  consequence  present  the  necessary  number 
of  values. 

For  example,  let  the  proposed  function  be 


f(x)  =  (x  -  a)m  \'x  -  b, 


m  being  a  positive  integer. 
Here  we  have 


,     ,    z m  (x  —  a)m 

f(x)  =  m  (x  -  a)m-1  Nx  -  b  +  , 

2  vx  -  b 

i mix  —  a)™-1      (x 

f"  (x)  =  m{m  -  i)  {x  -  a)™-2  Vx  -  b  +     v  ' — 


Vx-b  ${x-b)2 

Each  differentiation  causes  one  of  the  factors  of  (x  —  a)m  to  disappear  in  the 
first  term.  After  m  differentiations  these  factors  would  entirely  disappear;  and 
consequently  the  supposition  x  =  a,  in  causing  the  first  m-derived  functions  to 

vanish,  will  leave  the  radical  vx  —  b  to  remain  in  all  the  others. 


Conditions  of  Maxima  and  Minima  in  General.         445 


On  the  Conditions  foe  a  Maximum  oe,  Minimum  of  a  Function 

OF   ANT   ISUMBEK   OF  VaEIABLES   (Art.    1 63). 

The  conditions  for  a  maximum  or  a  minimum  in  the  case  of  two  or  of  three 
variables  have  been  given  in  Chapter  X. 

It  can  be  readily  seen  that  the  mode  of  investigation,  and  the  form  of  the 
conditions  there  given,  admit  of  extension  to  the  case  of  any  number  of  inde- 
pendent variables. 

"We  shall  commence  with  the  case  of  four  independent  variables.  Proceed- 
ing as  in  Art.  162,  it  is  obvious  that  the  problem  reduces  to  the  consideration  of 
a  quadratic  expression  in  four  variables  which  shall  preserve  the  same  sign  for 
all  real  values  of  the  variable. 

Let  the  quadratic  be  written  in  the  form 

#11  #12  +  #22#22  +  #33#32  +  #44#42  +  2^12^1  X%   +   20,\'zX\  X<$  +  2auX\X±    +   2#23#2#3, 


+   2«24#2#4  +  2  #34  XsXi, 


(I) 


in  which  «n,  #12,  022,  &c.,  represent  the  respective  second  differential  coefficients 
of  the  function,  as  in  Art.  162. 

We  shall  first  investigate  the  conditions  that  this  expression  shall  be  always 
a  positive  quantity ;  in  this  case  «n  evidently  is  necessarily  positive :  again, 
multiplying  by  an,  the  expression  may  be  written  in  the  following  form  : — 

(#1121  +  <?12#2  +  #13#3  +  #14£4)2  +  (#11  #22  -  #122)#22  +  (#11  #33  ~  #132)#32 

+  (#11#44  —  #142)#42  +  2(«ii023  —  #12#13)#2#3  +  2(#ii$24  —  #12#14)  #2#4 
+  2(#11#34  -  #13#14)#3#4.  "  (2) 

Also,  in  order  that  the  part  of  this  expression  after  the  first  term  shall  be 
always  positive,  we  must  have,  by  the  Article  referred  to,  the  following  condi- 
tions : — 


and 


#11  #22  -  #122  >  o, 

(#11#22  ~  #122)(#11#33  -  #132)  -  (#11  #23  ~  ^IJJ#13)2  >  O, 

#11  #22  —  #122,  #11  #23  —  #12  #13)  #11  #24  —  #12  #14 

#11#23  —  #12#13,  #11#33  —  #132,  #11  #34  —  #13#14 

#11  #24  —  #12  #14,         #11  #34  —  #13  #14,         #11  #44  —  #142 


>  O. 


(3) 

(4) 

(5) 


To  express  this  determinant  in  a  simpler  form,  we  write  it  as  follows  :- 

#12,                                    #13,  #14 

#11#22  —  #122,  #11  #23  —  #12  #13,  #11  #24  —  #12#14 

#11  #23  —  #12  #13,  #11#33  —  #133,  #11  #34  —  #13#14 

#11  #24  —  #12  #14,  #11  #34  —  #13  #14,  #11  #44  —  #142 


#11, 

I 

0, 

#11 

0, 

0, 

(6) 


446 


Notes. 


Next,  to  form  a  new  determinant,  multiply  the  first  row  by  012,  «i3,  #14,  suc- 
cessively, and  add  the  resulting  terms  to  the  2nd,  3rd,  and  4th  rows,  respec- 
tively ;  then,  since  each  term  in  the  rows  after  the  first  contains  an  as  a  factor, 
the  determinant  is  evidently  equivalent  to 


an" 


#11,  #12,  ai3,  #14 

#12,  #22,  #23,  #24 

#13,  #23}  #33,  #34 

#14,  #24,  #34,  #44 


(7) 


In  like  manner  the  relation  in  (4)  is  at  once  reducible  to  the  form 


#11 


#11,     #12,     #13 

#12,       #22,       #23 
#13,       #23,       #33 


>o. 


Hence  we  conclude  that  whenever  the  following  conditions  are  fulfilled, 
viz. : 


1 

#11,  #12 

#11  >  0, 

#12,  #22 

>°, 

#11,  #12, 

#13 

#12,  #22, 

#23 

>o, 

#13,   #23, 

#33 

#11,  #12,  #13,  #14 
#12,  #22,  #23,  #24 
#13,  #23,  #33,  #34 
#14,  #24,  #34,   #44 


>o, 


(8) 


the  quadratic  expression  (1)  is  positive  for  all  real  values  of  x\,  #2,  #3,  #4- 

Accordingly  the  conditions  are  the  same  as  in  the  case  (Art.  162)  of  three 
variables,  %\,  %%,  #3 ;  with  the  addition  that  the  determinant  (7)  shall  be  also 
positive. 

In  like  manner  it  can  be  readily  seen  that  if  the  second  and  fourth  of  the 
preceding  determinants  be  positive,  and  the  two  others  negative,  the  quadratic, 
expression  (1)  is  negative  for  all  real  values  of  the  variables. 

The  last  determinant  in  (8)  is  called  the  discriminant  of  the  quadratic  func- 
tion, and  the  preceding  determinant  is  derived  from  it  by  omitting  the  extreme 
row  and  column,  and  the  other  is  derived  from  that  in  like  manner. 

"When  the  discriminant  vanishes,  it  can  be  seen  without  difficulty  that  the 
expression  (1)  is  reducible  to  the  sum  of  three  squares. 

It  can  be  easily  proved  by  induction  that  the  preceding  principle  holds  in 
general,  and  that  in  the  case  of  n  variables  the  conditions  can  be  deduced  from 
the  discriminant  in  the  manner  indicated  above. 

According  as  the  number  of  rows  in  a  determinant  is  even  or  odd,  the  de- 
terminant is  said  to  be  one  of  an  even  or  of  an  odd  order. 


Conditions  of  Maxima  and  Minima  in  General.         447 

If  the  notation  already  adopted  be  generalized,  the  coefficient  of  xr2  is  de- 
noted by  cirr,  and  that  of  xrxm,  by  2arm.  In  this  case  the  discriminant  of  the 
quadratic  function  in  n  variables  is 


#11,  #12,  #13, 

#12,  #22,  #23, 

#13,  #23,  #33, 

#lra»  #2m,  #3h, 


#1? 


#3» 


(Inn 


(9) 


and  the  conditions  that  the  quadratic  expression  shall  be  always  positive  are, 
that  the  determinant  (9)  and  the  series  of  determinants  derived  in  succession  by 
erasing  the  outside  row  and  column  shall  be  all  positive. 

To  establish  this  result,  we  multiply  the  quadratic  function  by  #n,  and  it  is 
evident  that  it  may  be  written  in  the  form 

(#11#1  +  #12#2  +  •  •  •  #ln#n)2  +  (#11#22  -  #122)  #22  +  •   .  .  +  (#ll#nn  —  #1m2)  Xn% 
+  2  (#n#23  —  #12#13)#2#3  +  &C.  -f  (2#n#m  —  a\rCi\n)xrXn  +  •  •  • 

In  order  that  this  should  be  always  positive,  it  is  necessary  that  the  part 
after  the  first  term  should  be  always  positive.  This  is  a  quadratic  function  of 
the  n  —  1  variables  x%,  %%,  .  .  .  xn.  Accordingly,  assuming  that  the  conditions  in 
question  hold  for  it,  its  discriminant  must  be  positive,  as  also  the  series  of  deter- 
minants derived  from  it.     But  the  discriminant  is 


#11  #22   -  #12  ,  #11  #23  —  #12  #13, 

#11  #23  —  #12  #13,       #11  #33  —  #132, 
#11  #24  —  #12  #14,       #11  #34  —  #13  #14, 


#1 1  #2w  —  #12  #1m,       #11  #3m  —  #13  #1«, 


#ll#2w  —  #12#l»i 
#ll#3w  —  #13#lw 
#ll#4w  —  #14#lrc 


#11  #««—  #in2 


(10) 


"Writing  this  as  in  (6),  and  proceeding  as  before,  it  is  easily  seen  that  the 
determinant  becomes 


011* 


#11) 

#12, 

#13, 

•    •       #lw 

#12, 

#22, 

#23, 

•    •      #2w 

#13) 

#23, 

#33)       • 

•    •       #3» 

#l?l) 

#2m, 

#3M)       . 

•     •        #WM 

(») 


i.  e.  the  discriminant  of  the  function  multiplied  by  #nn_3 


448 


Notes. 


Hence  we  infer,  that  if  the  principle  in  question  hold  for  n  —  I  variables  it 
holds  for  n.  But  it  has  been  shown  to  hold  in  the  cases  of  3  and  4  variables  ; 
consequently  it  holds  for  any  number. 

"We  conclude  finally  that  the  quadratic  expression  in  n  variables  is  always 
positive,  whenever  the  series  of  determinants 


011, 


011, 

012,      - 

•        •       01» 

011, 

012, 

013 

012, 

022, 

•       -       02w 

011, 

012 

t 

012, 

022, 

023 

,      • 

. 

012, 

022 

013, 

023, 

033 

• 

• 

• 

01n, 

02»,      • 

0?m 

,       (12) 


are  all  positive. 

Again,  if  the  series  of  determinants  of  an  even  order  be  all  positive,  and  those 
of  an  odd  order,  commencing  with  an,  be  all  negative,  the  quadratic  expression 
is  negative  for  all  real  values  of  the  variables. 

Hence  we  infer  that  the  number  of  independent  conditions  for  a  maximum 
or  a  minimum  in  the  case  of  n  variables  is  n  —  1,  as  stated  in  Art.  163. 

It  is  scarcely  necessary  to  state  that  similar  results  hold  if  we  interchange 
any  two  of  the  suffix  numbers  ;  i.e.  if  any  of  the  coefficients,  #22,  #33,  .  .  ann, 
be  taken  instead  of  an  as  the  leading  term  in  the  series  of  determinants. 

If  the  determinants  in  (12)  be  denoted  byAi,  A2,  A3,  .  .  .  An,  it  can  be  proved 
without  difficulty  that,  whenever  none  of  these  determinants  vanishes,  the  qua- 
dratic expression  under  consideration  may  be  written  in  the  form 


Ai  Ui*  +  -  w  +  -  U£  +  .  .  .  +  —  TTJ 

Ai  A2  A,i-i 


(13) 


Hence,  in  general,  when  the  quadratic  is  transformed  into  a  sum  of  squares, 
the  number  of  positive  squares  in  the  sum  depends  on  the  number  of  continua- 
tions of  signs  in  the  series  of  determinants  in  (12). 

It  is  easy  to  see  independently  that  the  series  of  conditions  in  (12)  are  neces- 
sary in  order  that  the  quadratic  function  under  consideration  should  be  always 
positive ;  the  preceding  investigation  proves,  however,  that  they  are  not  only 
necessary,  but  that  they  are  sufficient. 

Again,  since  these  results  hold  if  any  two  or  more  of  the  suffix  numbers  be 
interchanged,  we  get  the  following  theorem  in  the  theory  of  numbers :  that  if 
the  series  of  determinants  given  in  (12)  be  all  positive,  then  every  determinant 
obtained  from  them  by  an  interchange  of  the  suffix  numbers  is  also  necessarily 
positive. 

Also,  since,  when  a  quadratic  expression  is  reduced  to  a  sum  of  squares,  the 
number  of  positive  and  negative  squares  in  the  sum  is  fixed  (Salmon's  Higher 
Algebra,  Art.  162),  we  infer  that  the  number  of- variations  of  sign  in  any  series 
of  determinants  obtained  from  (12)  by  altering  the  suffix  numbers  is  the  same 
as  the  number  of  variations  of  sign  in  the  series  (12). 

As  already  stated,  a  quadratic  expression  can  be  transformed  in  an  infinite 
number  of  ways  by  linear  transformations  into  the  sum  of  a  number  of  squares 
multiplied  by  constant  coefficients  ;  there  is,  however,  one  mode  that  is  unique, 
viz.,  what  is  styled  the  orthogonal  transformation. 


Conditions  of  Maxima  and  Minima  in  General.        449 

In  this  case,  if  Xi,  X2,  Z3,  .  .  .  Xn  denote  the  new  linear  functions,  we 
have 

V=  xi*  +  xz2  +  .  .  •  +  Xr?  =  X13  +  Z22  +  &c.  +  XM2 ; 

and  also,  denoting  the  coefficients  of  the  squares  in  the  transformed  expression 
by  01,  «2,  •  •  •  0w, 

V  =  a\\X^  +  022#22  +  •    •    •  +  Onn&n*  +  •    •    •  +  2a\%X\%z  +  2air%l%r  +  .    .    . 

=  «iZi2  +  «2-^22  +  .    .    .  thiXn2. 

Hence,  equating  the  discriminants  of  U  -  \V for  the  two  systems,  we  get 


011  —  A, 

012,            .    . 

01m 

012, 

022  —  A, 

02w 

«13) 

023,             •     • 

03» 

017 


02m  1 


0MM  ~  A 


=  (01  -  A)  (a3  -  A)  .  .  .  (an  -  A).      (14) 


Accordingly,  the  coefficients  01,  02,  .  .  .  an,  are  the  roots  of  the  determinant 
at  the  left-hand  side  of  equation  (14). 

Moreover,  in  order  that  the  function  U  should  be  always  positive  or  always 
negative  for  all  real  values  of  the  variables  x\,  %%,...  xn,  the  coefficients 
«i,  02,  •  •  •  0«j  must  be  all  positive  in  the  former  case,  and  all  negative  in  the 
latter;  and  consequently,  in  either  case,  the  roots  of  the  determinant  in  (14) 
must  all  have  the  same  sign. 

The  application  of  this  result  to  the  determination  of  the  conditions  of 
maxima  and  minima  is  easily  seen  ;  however,  as  the  conditions  thus  arrived  at 
are  clumsy  and  complicated  in  comparison  with  those  given  in  (12),  it  is  not  con- 
sidered necessary  to  enter  into  their  discussion  here. 


2  G 


INDEX. 


AcNODE,  259. 

Approximations,  42. 

further  trigonometrical  applica- 
tions of,  130-8. 
Arbogast's  method  of  derivations,  88. 
Arc  of  plane   curve,  differential  ex- 
pressions for,  220,  223. 
Archimedes,  spiral  of,  301,  303. 
Asymptotes,  definition  of,  242,  249. 

method  of  finding,  242,  245. 

number  of,  243. 

parallel,  247. 

of  cubic,  249,^  325. 

in  polar  coordinates,  250. 

circular,  252. 

Bernoulli's  numbers,  93. 

series,  70. 
Bertrand,  on  limits  of  Taylor's  series, 

77- 

Bobillier's  theorem,  368,  374. 

Boole,  on   transformation  of   coordi- 
nates, 412. 

Brigg's  logarithmic  system,  26. 

Burnside,  on  covariants,  412. 

Cardioid,  297,  372. 

Cartesian    oval,    or    Cartesian,    233, 

375- 
third  focus,  376. 
tangent  to,  379. 

confocals  intersect  orthogonally, 
381. 
Casey,    on  new  form   of  tangential 
equation,  339. 
on  cycloid,  373. 
on  Cartesians,  382. 
Cassini,  oval  of,  233,  333. 
Catenary,  288,  321. 
Cayley,  259,  266. 
Centre  of  curve,  237. 


Centrode,  363. 

Change  of  single  independent  variable, 

399- 
of  two  independent  variables,  403 , 
410. 
Chasles,  on  envelope  of  a  carried  right 
line,  356. 
construction  for  centre  of  instan- 
taneous rotation,  359. 
generalization  of  method  of  draw- 
ing normals  to  a  roulette,  360. 
on  epicycloids,  373. 
on  Cartesian  oval,  376. 
on  cubics,  418. 
Circle  of  inflexions  in  motion  of  a  plane 

area,  354,  358,  367,  374. 
Conchoid  of  Mcomedes,  332,  361. 

centre  of  curvature  of,  370. 
Concomitant  functions,  411. 
Condition  that  Pdx  +  Qdy  is  a  total 

differential,  146. 
Conjugate  points,  259. 
Contact,  different  orders  of,  304. 
Convexity  and  concavity,  278. 
Crofton,  on  Cartesian  oval,  378,  379, 

380. 
Crunode,  259. 

Cubics,  262,  281,  323,  334. 
Curvature,  radius  of,  286/  287,  295, 
297,  301. 

chord  of,  296. 

at  a  double  point,  310. 

at  a  cusp,  311,  313. 

measure  of,  on  a  surface,  209. 
Cusps,  259,  266,  315. 

curvature  at,  311. 
Cycloid,  335,  356. 

equation  of,  335,  336. 

radius  of  curvature,  and  evolute, 

337-     ' 
length  of  arc,  338. 


452 


Index. 


Descartes,  on  normal  to  a  roulette,  336. 

ovals  of,  375. 
Differential  coefficients,  definition,  5. 

successive,  34. 
Differentiation,  of  a  product,  13,  14. 

a  quotient,  15. 

a  power,  16,  17. 

a  function  of  a  function,  17. 

an  inverse  function,  18. 

trigonometrical  functions,  19,  20. 

circular  functions,  21,  22. 

logarithm,  25. 

exponential  functions,  26. 

functions  of  two  variables,  115. 
three  or  more  variables,  117. 

an  implicit  function,  120. 

partial,  113,  406. 

of  a  function  of  two  variables, 

US- 
of  three  or  more  variables, 

applications  in  plane  trigono- 
metry, 130. 

in  spherical  trigonome- 
try, 133- 
successive,  144. 
of  <p  (x  +  at,  y  4-  fit)  with  respect 
to  t,  148. 
Discriminant  of  a  ternary   quadratic 
expression,  129,  194,  196. 
of  any  quadric,  447. 
Double  points,  258,  261. 

Elimination,  of  constants,  384. 

of  transcendental  functions,  386. 

of  arbitrary  functions,  387,  396. 
Envelope,  270. 

of  La2  +  2Ma  +  JV=  O,  272. 

of  a  system  of  confocal  conies, 
Ex.  8,  p.  276. 

of  a  carried  curve,  355. 

centre  of  curvature  of,  357. 
Epicyclics,363. 

are  epi-  or  hypo -trochoids,  366. 
Epicycloids   and    hypocycloids,   339, 

356. 
radius  of  curvature  of,  342. 
cusps  in,  341. 
double  generation  of,  343. 
evolute  of,  344. 
length  of  arc,  345. 
pedal,  346,  372. 
regarded  as  envelope,  347. 


Epitrochoids  and  hypotrochoids,  347. 

ellipse  as  a  case  of,  348,  363. 

centre  of  curvature  of,  351. 

double  generation  of,  367. 
Equation  of  tangent  to  a  plane  curve, 
212,  218. 

normal,  215. 
Errors  in  trigonometrical  observation, 

135. 
Euler,  formulae  for  sin  x  and  cos  x,  69. 

theorem  on  homogeneous   func- 
tions, 123,  127,  148,  162. 

on  double  generation  of  epi-  and 
hypocycloids,  344. 
Evolute,  297. 

of  parabola,  298. 

of  ellipse,  299,  308 ;  as  an  enve- 
lope, 297. 

of  equiangular  spiral,  300. 
Expansion  of  a  function  by  Taylor's 
series,  61. 

of  (p(x  +  h,  y  +  k),  156. 

of  <p(x  +  h,  y  +  k,  z  +  I),  159. 

Family  of  curves,  270. 

Ferrers,  on  Bobillier's  theorem,  369. 

on  Steiner's  envelope,  442. 
Folium  of  Descartes,  333. 
Functions,  elementary  forms  of,  2. 

continuous,  3. 

derived,  3. 

successive,  34. 

examples  of,  46. 

partial  derived,  113. 

elliptic,  illustrations  of,  136,  138. 

Graves,  on  a  new  form  of  tangential 
equation,  339. 

Harmonic  polar  of  point  of  inflexion 

on  a  cubic,  281. 
Huygens,  approximation  to  length  of 

circular  arc,  66. 
Hyperbolic  branches  of  a  curve,  246. 
Hypocycloid,  see  epicycloid. 
Hypotrochoid,  see  epitrochoid. 

Indeterminate  forms,  96. 

treated  algebraically,  96-9. 

treated  by  the  calculus,  99,  et  seq. 
Infinitesimals,  orders  of,  36. 

geometrical  illustration,  57. 
Inflexion,  points  of,  279,  281. 

in  polar  coordinates,  303. 


Index. 


453 


Intrinsic  equation  of  a  curve,  304. 

of  a  cycloid,  338. 

of  an  epicycloid,  350. 

of  the  involute  of  a  circle,  301. 
Inverse  curves,  225. 

tangent  to,  225. 

radius  of  curvature,  295. 

conjugate  Cartesians  as,  378. 
Involute,  297. 

of  circle,  300,  358,  374. 
-  of  cycloid,  356. 

of  epicycloid,  357. 

Jacobians,  415-27. 

Lagrange,   on  derived  functions,    4, 
note. 
on  limits  of  Taylor's  series,  76. 
on  addition  of  elliptic  integrals, 

136. 
theorem  on  expansion  in  series, 

on  Euler's  theorem,  163. 
condition  for  maxima  and  minima, 
.  I91*.  197,  i?9,  202. 
La  Hire,  circle  of  inflexions,  354. 

on  cycloid,  373. 
Landen's  transformation    in    elliptic 

functions,  133. 
Laplace's  theorem  on   expansion  in 

series,  154. 
Legendre,  on  elliptic  functions,  137. 
on  rectification  of  curves,  233. 
Leibnitz,  on  the  fundamental  principle 
of  the  calculus,  40. 
theorem  on  the  nth  derived  func- 
tion of  a  product,  51. 
on  tangents  to  curves  in  vectorial 
coordinates,  234. 
Lemniscate,  259,  277,  296,  329,  333. 
Limacon,  is  inverse  to  a  conic,  227, 

.  .  33i,  334.  349,  361,  372. 
Limiting  ratios,  algebraic  illustration 

.of>  5- 
trigonometrical  illustration,  7. 

Limits,  fundamental  principles  as  to, 

11. 

Maclaurin,  series,  65,  81. 

on  harmonic  polar  for  a  cubic,  282. 
Mannheim,  construction  for  axes  of  an 

ellipse^  374. 
Maxima  or  minima,  164. 

geometrical  examples,  164,  183. 


algebraic  examples,  166. 

„     ax2  +  2bxy  +  a/2 

of  -r-= ^ ~,  166,  177. 

ax2  +  2b' xy  +  cy2' 

condition  for,  169,  174. 

problem  on  area  of  section  of  a 
right  cone,  181. 

for  implicit  functions,  185. 

quadrilateral  of  given  sides,  186. 

for    two    variables,     191  ;     La- 
grange's condition,   191,   197. 

for  functions  of  three  variables, 
198. 
of  n  variables,  199,  447. 

application  to  surfaces,  200. 

undetermined  multipliers  applied 
to,  204. 
Multiple  points  on  curves,  256,  265, 

367. 
Multipliers,  method  of  undetermined, 
204. 

Napier,  logarithmic  system,  25. 
Navier,    geometrical    illustration    of 
fundamental  principles  of  the 
calculus,  8. 
on  Taylor's  theorem,  443. 
Newton's  definition  of  fluxion,  10. 
prime  and  ultimate  ratios,  40. 
expansions  of  sin  x,  cos  x,  sin"1  x, 
&c,  64,  69. 

by  differential  equations,  85. 
method  of  investigating  radius  of 

curvature,  291. 
on  evolute  of  epicycloid,  345. 
Nicomedes,  conchoid  of,  332. 
Node,  259. 
Normal,  equation  of,  215. 

number  passing  through  a  given 

point,  220. 
in  vectorial  coordinates,  233 . 

Orthogonal  transformations,  409,  414, 

449. 
Osc-node,  259. 
Osculating  curves,  309. 

circle,  291,  306. 

conic,  317. 
Oscul-inflexion,  point  of,  314,  317. 

Parabola,  of  the  third  degree,  262,  288. 

osculating,  318. 
Parabolic  branches  of  a  curve,  246. 
Parameter,  270. 


454 


Index. 


Partial  differentiation,  113,  406. 
Pascal,  limacon  of,  227. 
Pedal,  227. 

tangent  to,  227. 

examples  of,  230. 

negative,  227. 
Pliicker,  on  locus  of  cusps  of  cubics 

having  given  asymptotes,  265. 
Points,  de  rebroussement,  266. 

of  inflexion,  279. 
Polar  conic  of  a  point,  219. 
Proctor,  definition  of  epi-  and  hypo- 
cycloids,  399. 

epicyclics,  366. 
Ptolemy,  epicyclics,  366. 

Quetelet,  on  Cartesian  oval,  376,  381. 

Radius  of  curvature,  286. 

in    Cartesian    coordinates,    287, 

289. 
in  r,  p  coordinates,  295. 
in  polar  coordinates,  301. 
at  singular  points,  310. 
of  envelope   of   a  moving   right 
line,  358. 
Eeauleaux,  on  centrodes  of  moving 

areas,  363. 
Eeciprocal  polars,  228,  230. 
Remainder  in  series,  Taylor's,  76,  79. 

Maclaurin's,  81. 
Resultant  of  concurrent  lines,  234. 
Roberts,  W.,  extension  of  method  of 

inversion,  429. 
Rotation,  of  a  plane  area,  359. 

centre  of  instantaneous,  360,  364. 
of  a  rigid  body,  371. 
Roulettes,  335. 

normal  to,  336. 

centre  of  curvature,   352  ;    Sa- 

vary's  construction,  352. 
circle  of  inflexions  of,  354. 
motion  of  a  plane  figure  reduced 

to,  362. 
spherical,  370. 

Savary's  construction  for  centre   of 

curvature  of  roulette,  353. 
Series,  Taylor's,  61,  70,  76. 

binomial,  63,  82. 

logarithmic,  63,  82. 

for  sin#  and  cos;r,  64,  66,  81. 

Maclaurin's,  64,  81. 


exponential,  65,  81. 

Bernoulli's,  70. 

convergent  and  divergent,  72,  75. 

for  sin-1  as,  68,  85. 

for  tan~1.r,  68,  84. 

for  sin  mas  and  cos  mx,  87. 

Arbogast's,  88. 

Lagrange's,  151. 
Spinode,  259. 
Stationary,  points,  266. 

tangents,  282. 
Subtangent  and  subnormal,  215. 

polar,  223. 
Symbols,  separation  of,  53. 

representation  of  Taylor's  theo- 
rem by,  70,  160. 

Tacnode,  266. 

Tangent  to  curve,  212,  218,  258. 

number  through  a  point,  219. 

expression  for  perpendicular  on, 
217,  224. 

expression  for  intercept  on,  232. 
Taylor's  series,  61. 

symbolic  form  of,  70. 

Lagrange  on  limits  of,  j6. 

extension  to  two  variables,  1560 
to  three  variables,  159. 

symbolic  form  of,  160. 

on  inapplicability  of,  443. 
Three-cusped  hypocycloid,  350,  372, 

430,  442. 
Tracing  of  curves,  322,  328. 
Transformations,  linear,  408. 

orthogonal,  409,  449. 
Trisectrix,  332. 
Trochoids,  339. 

Ultimate  intersection,  locus  of,  271. 

for  consecutive  normals,  290. 
Undetermined  multipliers,  application 
to  maxima  and  minima,  204. 

applied  to  envelope,  273. 
Undulation,  points  of,  280. 

Variables,  dependent  and  indepen- 
dent, 1. 

Variations  of  elements  of  a  triangle, 
plane,  130;  spherical,  133. 

Vectorial  coordinates,  233. 

Whewell,  on  intrinsic  equation,  304. 


THE  END. 


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